32:
89:
726:
but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value. Both proofs involved what is known today as the
6178:
6049:
9996:
9670:
10029:
A real-valued function is upper as well as lower semi-continuous, if and only if it is continuous in the usual sense. Hence these two theorems imply the boundedness theorem and the extreme value theorem.
417:
1402:
1341:
588:
5942:
5712:
2624:
2953:
2533:
5774:
1281:
2062:
2002:
859:
9768:
and the theorem is true. In all other cases, the proof is a slight modification of the proofs given above. In the proof of the boundedness theorem, the upper semi-continuity of
1890:
955:
8847:
7933:
7323:
1565:
9428:
9298:
8526:
8396:
4676:
4169:
3604:
8663:
3018:
2816:
1729:
9739:
2482:
1697:
8779:
7865:
7255:
4616:
4109:
3544:
2154:
We first prove the boundedness theorem, which is a step in the proof of the extreme value theorem. The basic steps involved in the proof of the extreme value theorem are:
1183:
4848:
4347:
1245:
891:
807:
1662:
1520:
739:
The following examples show why the function domain must be closed and bounded in order for the theorem to apply. Each fails to attain a maximum on the given interval.
9198:
9083:
8296:
8181:
7727:
6822:
5240:
5186:
2659:
2440:
9766:
7768:
6863:
3042:
1942:
1636:
1494:
1030:
9913:
9587:
6550:
9144:
8631:
8242:
7798:
7689:
7614:
7031:
7005:
6976:
6787:
4043:
4014:
3752:
3717:
3375:
1121:
775:
9380:
9325:
8959:
8478:
8423:
8051:
7435:
5844:
1604:
1462:
1062:
987:
6669:
6353:
6254:
3349:
3288:
3207:
2845:
2246:
1758:
119:
9353:
8712:
8451:
4550:
2125:
9500:
9448:
9218:
9103:
9057:
9037:
9017:
8925:
8867:
8753:
8733:
8683:
8546:
8316:
8201:
8155:
8135:
8115:
8017:
7953:
7839:
7819:
7654:
7634:
7588:
7536:
7516:
7493:
7401:
7343:
7229:
7209:
7186:
7166:
7146:
7111:
7091:
7071:
7051:
6903:
6883:
6749:
6729:
6709:
6689:
6640:
6605:
6585:
6492:
6472:
6437:
4958:
4868:
4814:
4794:
4774:
4754:
4696:
4590:
4570:
4521:
4463:
4367:
4313:
4293:
4273:
4253:
4189:
4083:
4063:
3988:
3968:
3916:
3896:
3876:
3853:
3833:
3810:
3682:
3624:
3518:
3498:
3475:
3455:
3435:
3415:
3395:
3259:
3146:
3126:
3063:
2973:
2889:
2865:
2771:
2751:
2731:
2679:
2395:
2343:
2316:
2281:
2187:
2145:
2102:
1910:
1856:
1826:
1798:
1778:
1422:
1215:
1082:
911:
685:
650:
630:
610:
519:
499:
443:
298:
278:
250:
186:
9480:
9250:
8997:
8905:
8584:
8348:
8095:
7997:
7568:
7473:
7381:
6935:
6524:
6417:
6385:
6318:
6286:
4990:
4938:
4900:
4734:
4501:
4443:
4399:
4233:
3948:
3790:
3662:
3320:
3239:
3178:
3095:
2711:
2565:
2375:
1153:
475:
330:
230:
151:
11180:
6061:
11168:
5947:
335:
9523:, then the corresponding half of the boundedness theorem and the extreme value theorem hold and the values –∞ or +∞, respectively, from the
11290:
11175:
524:
1428:
asserts that a subset of the real line is compact if and only if it is both closed and bounded. Correspondingly, a metric space has the
11158:
11153:
11163:
11148:
10262:
10450:
11143:
17:
5725:
9802:) < ∞, but that is enough to obtain the contradiction. In the proof of the extreme value theorem, upper semi-continuity of
10760:
10514:
1286:
1346:
11321:
10312:
5868:
5638:
11258:
11117:
10177:
10147:
10118:
10088:
2574:
75:
53:
46:
10672:
10588:
2894:
11253:
11185:
10810:
10665:
10633:
10392:
5151:
2568:
2490:
728:
10886:
10863:
10578:
10206:
193:
9859:
9533:
1250:
11316:
10976:
10914:
10709:
10583:
10255:
2007:
1947:
10462:
10440:
814:
11285:
2069:
11270:
11036:
10650:
10472:
1861:
918:
8784:
7870:
7260:
1525:
10655:
10425:
5347:
9385:
9255:
8483:
8353:
5491:
is not bounded. Since every continuous function on is bounded, this contradicts the conclusion that
4621:
4114:
3549:
11074:
11021:
9524:
3021:
10482:
8636:
2978:
2776:
1706:
11190:
10961:
10509:
10248:
9706:
2445:
1667:
1567:
is also open. Given these definitions, continuous functions can be shown to preserve compactness:
1429:
40:
8758:
7844:
7234:
4595:
4088:
3523:
1166:
10956:
10628:
4819:
4318:
2868:
1425:
1223:
864:
780:
1641:
1499:
11084:
10966:
10787:
10735:
10541:
10519:
10387:
9149:
9062:
8247:
8160:
7697:
6792:
5781:
5211:
5157:
2629:
2400:
166:
57:
9748:
7732:
6827:
3024:
1915:
1609:
1467:
994:
11210:
11069:
10981:
10638:
10573:
10536:
10457:
10445:
10430:
10402:
10165:
10161:
6529:
5540:
10105:
9108:
8589:
8206:
7777:
7659:
7593:
7010:
6984:
6940:
6757:
4022:
3993:
3722:
3687:
3354:
1435:
The concept of a continuous function can likewise be generalized. Given topological spaces
1091:
745:
11026:
10645:
10492:
9358:
9303:
8930:
8456:
8401:
8022:
7406:
5791:
2197:
1805:
1801:
1580:
1438:
1035:
960:
10235:
6645:
6329:
6230:
3325:
3264:
3183:
2821:
2222:
1734:
95:
8:
11046:
10971:
10858:
10815:
10566:
10551:
10382:
10370:
10357:
10317:
10297:
10217:
10202:
9330:
8691:
8428:
4529:
2162:
189:
9856:
proves a similar result for the infimums of lower semicontinuous functions. A function
2107:
11135:
11110:
10941:
10894:
10835:
10800:
10795:
10775:
10770:
10765:
10730:
10677:
10660:
10561:
10435:
10420:
10365:
10332:
9485:
9433:
9203:
9088:
9042:
9022:
9002:
8910:
8852:
8738:
8718:
8668:
8531:
8301:
8186:
8140:
8120:
8100:
8002:
7938:
7824:
7804:
7639:
7619:
7573:
7521:
7501:
7478:
7386:
7328:
7214:
7194:
7171:
7151:
7116:
7096:
7076:
7056:
7036:
6888:
6868:
6734:
6714:
6694:
6674:
6610:
6590:
6555:
6477:
6442:
6422:
5614:
4943:
4853:
4799:
4779:
4759:
4739:
4681:
4575:
4555:
4506:
4448:
4352:
4298:
4278:
4258:
4238:
4174:
4068:
4048:
3973:
3953:
3901:
3881:
3861:
3838:
3818:
3795:
3667:
3609:
3503:
3483:
3460:
3440:
3420:
3400:
3380:
3244:
3131:
3111:
3048:
2958:
2874:
2850:
2756:
2736:
2716:
2664:
2380:
2328:
2286:
2251:
2172:
2130:
2087:
1895:
1841:
1811:
1783:
1763:
1407:
1200:
1190:
1067:
896:
691:
655:
635:
615:
595:
504:
484:
428:
283:
263:
235:
171:
9453:
9223:
8964:
8872:
8551:
8321:
8056:
7958:
7541:
7440:
7348:
6908:
6497:
6390:
6358:
6291:
6259:
5594:* defined on the hyperreals between 0 and 1. Note that in the standard setting (when
4963:
4905:
4873:
4701:
4468:
4404:
4372:
4194:
3921:
3757:
3629:
3293:
3212:
3151:
3068:
2684:
2538:
2348:
2068:
Slightly more generally, this is also true for an upper semicontinuous function. (see
1126:
448:
303:
203:
124:
11275:
11099:
11031:
10853:
10830:
10704:
10697:
10600:
10415:
10307:
10214:
10173:
10157:
10143:
10114:
10084:
5343:
5013:
11233:
11016:
10929:
10909:
10840:
10750:
10692:
10684:
10618:
10531:
10292:
10287:
10056:
2203:
Use continuity to show that the image of the subsequence converges to the supremum.
695:
478:
2147:
follows. Also note that everything in the proof is done within the context of the
11295:
11280:
11064:
10919:
10899:
10868:
10845:
10825:
10719:
10375:
10322:
10193:
9520:
5551:. The interval has a natural hyperreal extension. Consider its partition into
1217:
is said to be compact if it has the following property: from every collection of
719:
6173:{\displaystyle \mathbf {st} (f^{*}(x_{i_{0}}))=f(\mathbf {st} (x_{i_{0}}))=f(c)}
11205:
11104:
10951:
10904:
10805:
10608:
10205:
by
Jacqueline Wandzura with additional contributions by Stephen Wandzura, the
9777:
6044:{\displaystyle \mathbf {st} (f^{*}(x_{i_{0}}))\geq \mathbf {st} (f^{*}(x_{i}))}
10623:
11310:
11079:
10934:
10820:
10524:
10499:
10060:
5556:
1194:
1186:
699:
16:
This article is about the calculus concept. For the statistical concept, see
10047:
Rusnock, Paul; Kerr-Lawson, Angus (2005). "Bolzano and
Uniform Continuity".
1828:. Thus, we have the following generalization of the extreme value theorem:
11089:
11059:
10924:
10487:
10231:
10197:
5548:
2148:
687:
which is what the extreme value theorem stipulates must also be the case.
10337:
10279:
10078:
2193:
2081:
707:
197:
23:
Continuous real function on a closed interval has a maximum and a minimum
1123:
in the last two examples shows that both theorems require continuity on
88:
11054:
10986:
10740:
10613:
10477:
10467:
10410:
6387:
and by the completeness property of the real numbers has a supremum in
11248:
10996:
10991:
10302:
10222:
2127:, the existence of the lower bound and the result for the minimum of
11243:
10745:
10271:
4756:
is bounded on this interval. But it follows from the supremacy of
4255:
is bounded on this interval. But it follows from the supremacy of
2485:
2166:
1218:
1193:, the appropriate generalization of a closed bounded interval is a
158:
9991:{\displaystyle \liminf _{y\to x}f(y)\geq f(x)\quad \forall x\in .}
9665:{\displaystyle \limsup _{y\to x}f(y)\leq f(x)\quad \forall x\in .}
1158:
698:, this theorem states that a continuous function from a non-empty
11094:
10347:
10022:
257:
253:
734:
11263:
10327:
9845:
9503:
6213:
5524:
5304:
4993:
3098:
703:
10212:
10342:
3835:
is an interval of non-zero length, closed at its left end by
421:
The extreme value theorem is more specific than the related
10240:
153:
showing the absolute max (red) and the absolute min (blue).
1404:. This is usually stated in short as "every open cover of
412:{\displaystyle f(c)\leq f(x)\leq f(d)\quad \forall x\in .}
9510:
5788:
lies in a suitable sub-interval of the partition, namely
10236:
http://mizar.org/version/current/html/weierstr.html#T15
1397:{\textstyle \bigcup _{i=1}^{n}U_{\alpha _{i}}\supset K}
1336:{\displaystyle U_{\alpha _{1}},\ldots ,U_{\alpha _{n}}}
5598: is finite), a point with the maximal value of
5507:
was continuous on . Therefore, there must be a point
5016:
of the real numbers, the least upper bound (supremum)
4999:
1349:
1253:
10113:. Boston: Prindle, Weber & Schmidt. p. 164.
9916:
9862:
9751:
9709:
9590:
9536:
9488:
9456:
9436:
9388:
9361:
9333:
9306:
9258:
9226:
9206:
9152:
9111:
9091:
9065:
9045:
9025:
9005:
8967:
8933:
8913:
8875:
8855:
8787:
8761:
8741:
8721:
8694:
8671:
8639:
8592:
8554:
8534:
8486:
8459:
8431:
8404:
8356:
8324:
8304:
8250:
8209:
8189:
8163:
8143:
8123:
8103:
8059:
8025:
8005:
7961:
7941:
7873:
7847:
7827:
7807:
7780:
7735:
7700:
7662:
7642:
7622:
7596:
7576:
7544:
7524:
7504:
7481:
7443:
7409:
7389:
7351:
7331:
7263:
7237:
7217:
7197:
7174:
7154:
7119:
7099:
7079:
7059:
7039:
7013:
6987:
6943:
6911:
6891:
6871:
6830:
6795:
6760:
6737:
6717:
6697:
6677:
6648:
6613:
6593:
6558:
6532:
6500:
6480:
6445:
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6393:
6361:
6332:
6294:
6262:
6233:
6064:
5950:
5871:
5794:
5728:
5641:
5214:
5160:
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4908:
4876:
4856:
4822:
4802:
4782:
4762:
4742:
4704:
4684:
4624:
4598:
4578:
4558:
4532:
4509:
4471:
4451:
4407:
4375:
4355:
4321:
4301:
4281:
4261:
4241:
4197:
4177:
4117:
4091:
4071:
4051:
4025:
3996:
3976:
3956:
3924:
3904:
3884:
3864:
3841:
3821:
3798:
3760:
3725:
3690:
3670:
3632:
3612:
3552:
3526:
3506:
3486:
3463:
3443:
3423:
3403:
3383:
3357:
3328:
3296:
3267:
3247:
3215:
3186:
3154:
3134:
3114:
3071:
3051:
3027:
2981:
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2897:
2877:
2853:
2824:
2779:
2759:
2739:
2719:
2687:
2667:
2632:
2577:
2541:
2493:
2448:
2403:
2383:
2351:
2331:
2289:
2254:
2225:
2175:
2133:
2110:
2090:
2010:
1950:
1918:
1898:
1864:
1844:
1814:
1786:
1766:
1737:
1709:
1670:
1644:
1612:
1583:
1528:
1502:
1470:
1441:
1410:
1289:
1226:
1203:
1169:
1129:
1094:
1070:
1038:
997:
963:
921:
899:
893:
is bounded but does not attain its least upper bound
867:
817:
783:
748:
658:
638:
618:
598:
527:
507:
487:
451:
431:
338:
306:
286:
266:
238:
206:
174:
127:
98:
10107:
Elementary
Calculus : An Infinitesimal Approach
9810:
implies that the limit superior of the subsequence {
583:{\displaystyle m\leq f(x)\leq M\quad \forall x\in .}
481:
on that interval; that is, there exist real numbers
260:, each at least once. That is, there exist numbers
7168:is a non-empty interval, closed at its left end by
2571:implies that there exists a convergent subsequence
2207:
1432:if every closed and bounded set is also compact.
1064:is bounded but never attains its least upper bound
718:The extreme value theorem was originally proven by
10046:
9990:
9907:
9760:
9733:
9664:
9581:
9494:
9474:
9442:
9422:
9374:
9347:
9319:
9292:
9244:
9212:
9192:
9138:
9097:
9077:
9051:
9031:
9011:
8991:
8953:
8919:
8899:
8861:
8841:
8773:
8747:
8727:
8706:
8677:
8657:
8625:
8578:
8540:
8520:
8472:
8445:
8417:
8390:
8342:
8310:
8290:
8236:
8195:
8175:
8149:
8129:
8109:
8089:
8045:
8011:
7991:
7947:
7927:
7859:
7833:
7813:
7792:
7762:
7721:
7683:
7648:
7628:
7608:
7582:
7562:
7530:
7510:
7487:
7467:
7429:
7395:
7375:
7337:
7317:
7249:
7223:
7203:
7180:
7160:
7140:
7105:
7085:
7065:
7045:
7025:
6999:
6970:
6929:
6897:
6877:
6857:
6816:
6781:
6743:
6723:
6703:
6683:
6663:
6634:
6599:
6579:
6544:
6518:
6486:
6466:
6431:
6411:
6379:
6347:
6312:
6280:
6248:
6172:
6043:
5936:
5838:
5768:
5706:
5234:
5180:
4984:
4952:
4932:
4894:
4862:
4842:
4808:
4788:
4768:
4748:
4728:
4690:
4670:
4610:
4584:
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4544:
4515:
4495:
4457:
4437:
4393:
4361:
4341:
4307:
4287:
4267:
4247:
4227:
4183:
4163:
4103:
4077:
4057:
4037:
4008:
3982:
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3942:
3910:
3890:
3870:
3847:
3827:
3804:
3784:
3746:
3711:
3676:
3656:
3618:
3598:
3538:
3512:
3492:
3469:
3449:
3429:
3409:
3389:
3369:
3343:
3314:
3282:
3253:
3233:
3201:
3172:
3140:
3120:
3089:
3057:
3036:
3012:
2967:
2947:
2883:
2859:
2839:
2810:
2765:
2745:
2725:
2705:
2673:
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2618:
2559:
2527:
2476:
2434:
2389:
2369:
2337:
2310:
2275:
2240:
2181:
2139:
2119:
2096:
2056:
1996:
1936:
1904:
1884:
1850:
1820:
1792:
1772:
1752:
1723:
1691:
1656:
1630:
1598:
1559:
1514:
1488:
1456:
1416:
1396:
1335:
1275:
1239:
1209:
1177:
1147:
1115:
1076:
1056:
1024:
981:
949:
905:
885:
853:
801:
769:
679:
644:
632:are necessarily the maximum and minimum values of
624:
604:
582:
513:
493:
469:
437:
411:
324:
292:
272:
244:
224:
180:
145:
113:
7636:is the point we are seeking i.e. the point where
5937:{\displaystyle f^{*}(x_{i_{0}})\geq f^{*}(x_{i})}
5707:{\displaystyle f^{*}(x_{i_{0}})\geq f^{*}(x_{i})}
5590: is also naturally extended to a function
425:, which states merely that a continuous function
11308:
9918:
9592:
2027:
1967:
2619:{\displaystyle (x_{n_{k}})_{k\in \mathbb {N} }}
1496:is said to be continuous if for every open set
1159:Generalization to metric and topological spaces
6219:
5311:Alternative Proof of the Extreme Value Theorem
10256:
10156:
10142:. Reading: Addison-Wesley. pp. 706–707.
8665:. This however contradicts the supremacy of
5530:
4503:. This however contradicts the supremacy of
2948:{\displaystyle f(x_{{n}_{k}})>n_{k}\geq k}
735:Functions to which the theorem does not apply
10166:"The Boundedness and Extreme–Value Theorems"
2104:. By applying these results to the function
9695:is bounded above and attains its supremum.
2528:{\displaystyle (x_{n})_{n\in \mathbb {N} }}
690:The extreme value theorem is used to prove
10263:
10249:
6789:then we are done. Suppose therefore that
5769:{\displaystyle c=\mathbf {st} (x_{i_{0}})}
5154:tells us that there exists a subsequence {
3377:is another point, then all points between
2070:compact space#Functions and compact spaces
1760:is closed and bounded for any compact set
11291:Regiomontanus' angle maximization problem
10083:. New York: McGraw Hill. pp. 89–90.
8999:. But it follows from the supremacy of
8097:. But it follows from the supremacy of
3457:is an interval closed at its left end by
2610:
2519:
1878:
1717:
1276:{\textstyle \bigcup U_{\alpha }\supset K}
1171:
76:Learn how and when to remove this message
11134:
9908:{\displaystyle f:\to [-\infty ,\infty )}
9582:{\displaystyle f:\to [-\infty ,\infty )}
7656:attains its supremum, or in other words
5024:exists. It is necessary to find a point
87:
39:This article includes a list of general
10639:Differentiating under the integral sign
10103:
6474:. It is clear that the restriction of
4776:that there exists a point belonging to
4275:that there exists a point belonging to
2075:
2057:{\displaystyle f(q)=\inf _{x\in K}f(x)}
1997:{\displaystyle f(p)=\sup _{x\in K}f(x)}
11309:
10172:. New York: Springer. pp. 71–73.
9832:), but this suffices to conclude that
9511:Extension to semi-continuous functions
7770:and consider the following two cases:
5012:is bounded from above, hence, by the
10515:Inverse functions and differentiation
10244:
10213:
10137:
10076:
9482:. This contradicts the supremacy of
2345:is not bounded above on the interval
854:{\displaystyle f(x)={\frac {x}{1+x}}}
10072:
10070:
25:
10080:Principles of Mathematical Analysis
9527:can be allowed as possible values.
7475:so that all these points belong to
5000:Proofs of the extreme value theorem
3792:so that all these points belong to
1885:{\displaystyle f:K\to \mathbb {R} }
950:{\displaystyle f(x)={\frac {1}{x}}}
13:
10313:Free variables and bound variables
10131:
9961:
9899:
9893:
9755:
9728:
9635:
9573:
9567:
9515:If the continuity of the function
8842:{\displaystyle |f(x)-f(s)|<d/2}
7928:{\displaystyle |f(x)-f(s)|<d/2}
7318:{\displaystyle |f(x)-f(a)|<d/2}
5426:However, to every positive number
5005:Proof of the Extreme Value Theorem
3031:
2377:. Then, for every natural number
1560:{\displaystyle f^{-1}(U)\subset V}
877:
793:
553:
382:
45:it lacks sufficient corresponding
14:
11333:
11118:The Method of Mechanical Theorems
10194:A Proof for extreme value theorem
10187:
10140:Calculus : A Complete Course
10067:
10021:is bounded below and attains its
5456:is the least upper bound. Hence,
2196:that converges to a point in the
1731:, then this theorem implies that
729:Bolzano–Weierstrass theorem
710:attains a maximum and a minimum.
10673:Partial fractions in integration
10589:Stochastic differential equation
9423:{\displaystyle f(x)\leq M-d_{2}}
9293:{\displaystyle f(x)\leq M-d_{1}}
8521:{\displaystyle f(x)\leq M-d_{2}}
8391:{\displaystyle f(x)\leq M-d_{1}}
7538:and has therefore a supremum in
7148:is monotonic increasing. Hence
6288:then it attains its supremum on
6125:
6122:
6069:
6066:
6005:
6002:
5955:
5952:
5739:
5736:
4671:{\displaystyle |f(x)-f(s)|<1}
4164:{\displaystyle |f(x)-f(s)|<1}
3599:{\displaystyle |f(x)-f(a)|<1}
2208:Proof of the boundedness theorem
30:
10811:Jacobian matrix and determinant
10666:Tangent half-angle substitution
10634:Fundamental theorem of calculus
10170:A First Course in Real Analysis
10017:is lower semi-continuous, then
9960:
9852:Applying this result to −
9745:in , then the supremum is also
9691:is upper semi-continuous, then
9634:
7590:. We see from the above that
6587:which is less than or equal to
5718: = 0, ...,
5613:, by induction. Hence, by the
5602:can always be chosen among the
1892:is a continuous function, then
1163:When moving from the real line
552:
381:
18:Fisher–Tippett–Gnedenko theorem
10887:Arithmetico-geometric sequence
10579:Ordinary differential equation
10207:Wolfram Demonstrations Project
10097:
10040:
9982:
9970:
9957:
9951:
9942:
9936:
9925:
9902:
9887:
9884:
9881:
9869:
9719:
9713:
9656:
9644:
9631:
9625:
9616:
9610:
9599:
9576:
9561:
9558:
9555:
9543:
9469:
9457:
9398:
9392:
9268:
9262:
9239:
9227:
9187:
9175:
9127:
9115:
8986:
8968:
8894:
8876:
8821:
8817:
8811:
8802:
8796:
8789:
8658:{\displaystyle s+\delta \in L}
8614:
8596:
8573:
8555:
8496:
8490:
8366:
8360:
8337:
8325:
8285:
8273:
8225:
8213:
8084:
8060:
7986:
7962:
7907:
7903:
7897:
7888:
7882:
7875:
7757:
7751:
7710:
7704:
7672:
7666:
7557:
7545:
7462:
7444:
7370:
7352:
7297:
7293:
7287:
7278:
7272:
7265:
7211:is continuous on the right at
7135:
7123:
6959:
6947:
6924:
6912:
6852:
6846:
6805:
6799:
6770:
6764:
6658:
6652:
6629:
6617:
6574:
6562:
6513:
6501:
6461:
6449:
6406:
6394:
6374:
6362:
6342:
6336:
6307:
6295:
6275:
6263:
6243:
6237:
6167:
6161:
6152:
6149:
6129:
6118:
6109:
6106:
6086:
6073:
6038:
6035:
6022:
6009:
5995:
5992:
5972:
5959:
5931:
5918:
5902:
5882:
5833:
5801:
5763:
5743:
5701:
5688:
5672:
5652:
4979:
4967:
4927:
4909:
4889:
4877:
4723:
4705:
4658:
4654:
4648:
4639:
4633:
4626:
4490:
4472:
4432:
4408:
4388:
4376:
4222:
4198:
4151:
4147:
4141:
4132:
4126:
4119:
3970:. From the non-zero length of
3937:
3925:
3779:
3761:
3735:
3729:
3700:
3694:
3651:
3633:
3586:
3582:
3576:
3567:
3561:
3554:
3500:is continuous on the right at
3338:
3332:
3309:
3297:
3277:
3271:
3228:
3216:
3196:
3190:
3167:
3155:
3084:
3072:
3045:, a contradiction. Therefore,
3013:{\displaystyle f(x_{{n}_{k}})}
3007:
2985:
2923:
2901:
2834:
2828:
2811:{\displaystyle f(x_{{n}_{k}})}
2805:
2783:
2700:
2688:
2648:
2633:
2599:
2578:
2554:
2542:
2508:
2494:
2465:
2452:
2429:
2417:
2364:
2352:
2302:
2290:
2267:
2255:
2235:
2229:
2158:Prove the boundedness theorem.
2051:
2045:
2020:
2014:
1991:
1985:
1960:
1954:
1874:
1858:is a nonempty compact set and
1808:on any (nonempty) compact set
1747:
1741:
1724:{\displaystyle W=\mathbb {R} }
1680:
1674:
1638:is a continuous function, and
1622:
1548:
1542:
1480:
1142:
1130:
1104:
1098:
1051:
1039:
1007:
1001:
976:
964:
931:
925:
880:
868:
827:
821:
796:
784:
758:
752:
671:
659:
574:
562:
543:
537:
464:
452:
403:
391:
378:
372:
363:
357:
348:
342:
319:
307:
219:
207:
140:
128:
108:
102:
1:
10710:Integro-differential equation
10584:Partial differential equation
10033:
9734:{\displaystyle f(x)=-\infty }
8735:is continuous on the left at
5342:is a bounded set. Hence, its
4572:is continuous on the left at
2818:converges to the real number
2477:{\displaystyle f(x_{n})>n}
2080:We look at the proof for the
1780:, which in turn implies that
1692:{\displaystyle f(K)\subset W}
165:states that if a real-valued
10270:
8774:{\displaystyle \delta >0}
7860:{\displaystyle \delta >0}
7250:{\displaystyle \delta >0}
6326:By the Boundedness Theorem,
6227: If
5132:. Therefore, the sequence {
5090:). This defines a sequence {
5008:By the boundedness theorem,
4611:{\displaystyle \delta >0}
4104:{\displaystyle \delta >0}
3539:{\displaystyle \delta >0}
2161:Find a sequence so that its
1424:has a finite subcover". The
1178:{\displaystyle \mathbb {R} }
7:
10864:Generalized Stokes' theorem
10651:Integration by substitution
10104:Keisler, H. Jerome (1986).
9019:that there exists a point,
8117:that there exists a point,
6220:Proof from first principles
5784:. An arbitrary real point
5722:. Consider the real point
5188:}, which converges to some
5152:Bolzano–Weierstrass theorem
4843:{\displaystyle s-\delta /2}
4816:say, which is greater than
4342:{\displaystyle s-\delta /2}
4315:say, which is greater than
2569:Bolzano–Weierstrass theorem
1912:is bounded and there exist
1240:{\displaystyle U_{\alpha }}
886:{\displaystyle [0,\infty )}
802:{\displaystyle [0,\infty )}
694:. In a formulation due to
10:
11338:
10393:(ε, δ)-definition of limit
7694:Suppose the contrary viz.
5617:, there is a hyperinteger
5531:Proof using the hyperreals
5348:least upper bound property
5064:. Therefore, there exists
5060:is not an upper bound for
1657:{\displaystyle K\subset V}
1515:{\displaystyle U\subset W}
989:is not bounded from above.
809:is not bounded from above.
713:
15:
11322:Theorems in real analysis
11286:Proof that 22/7 exceeds π
11223:
11201:
11127:
11075:Gottfried Wilhelm Leibniz
11045:
11022:e (mathematical constant)
11007:
10879:
10786:
10718:
10599:
10401:
10356:
10278:
10138:Adams, Robert A. (1995).
9525:extended real number line
9502:and completes the proof.
9193:{\displaystyle d_{1}=M-M}
9078:{\displaystyle s-\delta }
8291:{\displaystyle d_{1}=M-M}
8176:{\displaystyle s-\delta }
7722:{\displaystyle f(s)<M}
6817:{\displaystyle f(a)<M}
5350:of the real numbers. Let
5235:{\displaystyle d_{n_{k}}}
5181:{\displaystyle d_{n_{k}}}
2654:{\displaystyle ({x_{n}})}
2435:{\displaystyle x_{n}\in }
2192:Show that there exists a
1283:, a finite subcollection
11037:Stirling's approximation
10510:Implicit differentiation
10458:Rules of differentiation
10061:10.1016/j.hm.2004.11.003
9761:{\displaystyle -\infty }
9085:. By the definition of
8685:and completes the proof.
8183:. By the definition of
7763:{\displaystyle d=M-f(s)}
7053:then all points between
6858:{\displaystyle d=M-f(a)}
5563:, with partition points
5377:. If there is no point
5264:)} is a subsequence of {
5044:be a natural number. As
3037:{\displaystyle +\infty }
1937:{\displaystyle p,q\in K}
1631:{\displaystyle f:V\to W}
1606:are topological spaces,
1489:{\displaystyle f:V\to W}
1343:can be chosen such that
1025:{\displaystyle f(x)=1-x}
11271:Euler–Maclaurin formula
11176:trigonometric functions
10629:Constant of integration
10218:"Extreme Value Theorem"
9824:)} is bounded above by
9794:)} is bounded above by
6545:{\displaystyle x\leq b}
5547: be an infinite
5430:, there is always some
4526:We must therefore have
3950: ; let us call it
3261:is one such point, for
2869:sequentially continuous
2713:is closed, it contains
722:in the 1830s in a work
592:This does not say that
445:on the closed interval
121:on the closed interval
60:more precise citations.
11240:Differential geometry
11085:Infinitesimal calculus
10788:Multivariable calculus
10736:Directional derivative
10542:Second derivative test
10520:Logarithmic derivative
10493:General Leibniz's rule
10388:Order of approximation
10077:Rudin, Walter (1976).
9992:
9909:
9776:only implies that the
9762:
9735:
9666:
9583:
9496:
9476:
9444:
9424:
9376:
9349:
9321:
9294:
9246:
9214:
9194:
9140:
9139:{\displaystyle M<M}
9099:
9079:
9059:which is greater than
9053:
9033:
9013:
8993:
8955:
8921:
8901:
8863:
8843:
8775:
8749:
8729:
8708:
8679:
8659:
8627:
8626:{\displaystyle M<M}
8580:
8542:
8522:
8474:
8447:
8419:
8392:
8344:
8312:
8292:
8238:
8237:{\displaystyle M<M}
8197:
8177:
8157:which is greater than
8151:
8131:
8111:
8091:
8047:
8013:
7993:
7949:
7929:
7861:
7835:
7815:
7794:
7793:{\displaystyle s<b}
7764:
7723:
7685:
7684:{\displaystyle f(s)=M}
7650:
7630:
7610:
7609:{\displaystyle s>a}
7584:
7564:
7532:
7512:
7489:
7469:
7431:
7397:
7377:
7339:
7319:
7251:
7225:
7205:
7182:
7162:
7142:
7107:
7087:
7067:
7047:
7027:
7026:{\displaystyle e>a}
7001:
7000:{\displaystyle a\in L}
6972:
6971:{\displaystyle M<M}
6931:
6899:
6879:
6859:
6818:
6783:
6782:{\displaystyle f(a)=M}
6745:
6725:
6705:
6685:
6665:
6636:
6601:
6581:
6546:
6520:
6488:
6468:
6433:
6413:
6381:
6349:
6314:
6282:
6250:
6174:
6045:
5938:
5840:
5782:standard part function
5770:
5708:
5555:subintervals of equal
5236:
5182:
5101:is an upper bound for
4986:
4954:
4934:
4896:
4864:
4844:
4810:
4790:
4770:
4750:
4730:
4692:
4672:
4612:
4586:
4566:
4546:
4517:
4497:
4459:
4439:
4395:
4363:
4343:
4309:
4289:
4269:
4249:
4229:
4185:
4165:
4105:
4079:
4059:
4039:
4038:{\displaystyle s<b}
4010:
4009:{\displaystyle s>a}
3984:
3964:
3944:
3912:
3892:
3872:
3849:
3829:
3806:
3786:
3748:
3747:{\displaystyle f(a)+1}
3713:
3712:{\displaystyle f(a)-1}
3678:
3658:
3620:
3600:
3540:
3514:
3494:
3471:
3451:
3431:
3411:
3391:
3371:
3370:{\displaystyle e>a}
3345:
3316:
3284:
3255:
3235:
3203:
3174:
3142:
3122:
3091:
3059:
3038:
3014:
2969:
2949:
2885:
2861:
2841:
2812:
2767:
2747:
2727:
2707:
2675:
2661:. Denote its limit by
2655:
2620:
2561:
2529:
2478:
2436:
2391:
2371:
2339:
2312:
2283:then it is bounded on
2277:
2242:
2183:
2141:
2121:
2098:
2058:
1998:
1938:
1906:
1886:
1852:
1822:
1794:
1774:
1754:
1725:
1693:
1658:
1632:
1600:
1561:
1516:
1490:
1458:
1418:
1398:
1370:
1337:
1277:
1241:
1211:
1179:
1149:
1117:
1116:{\displaystyle f(0)=0}
1078:
1058:
1026:
983:
951:
907:
887:
855:
803:
771:
770:{\displaystyle f(x)=x}
681:
646:
626:
606:
584:
515:
495:
471:
439:
413:
326:
294:
274:
246:
226:
182:
154:
147:
115:
92:A continuous function
11159:logarithmic functions
11154:exponential functions
11070:Generality of algebra
10948:Tests of convergence
10574:Differential equation
10558:Further applications
10547:Extreme value theorem
10537:First derivative test
10431:Differential operator
10403:Differential calculus
10203:Extreme Value Theorem
9993:
9910:
9763:
9736:
9667:
9584:
9497:
9477:
9445:
9425:
9377:
9375:{\displaystyle d_{1}}
9350:
9327:to be the minimum of
9322:
9320:{\displaystyle d_{2}}
9295:
9247:
9215:
9195:
9141:
9100:
9080:
9054:
9034:
9014:
8994:
8956:
8954:{\displaystyle M-d/2}
8922:
8907:. This means that
8902:
8864:
8844:
8776:
8750:
8730:
8709:
8680:
8660:
8628:
8581:
8543:
8523:
8475:
8473:{\displaystyle d_{1}}
8448:
8425:to be the minimum of
8420:
8418:{\displaystyle d_{2}}
8393:
8345:
8313:
8293:
8239:
8198:
8178:
8152:
8132:
8112:
8092:
8048:
8046:{\displaystyle M-d/2}
8014:
7999:. This means that
7994:
7950:
7930:
7862:
7836:
7816:
7795:
7765:
7724:
7686:
7651:
7631:
7616:. We will show that
7611:
7585:
7565:
7533:
7513:
7490:
7470:
7432:
7430:{\displaystyle M-d/2}
7398:
7378:
7340:
7320:
7252:
7231:, hence there exists
7226:
7206:
7183:
7163:
7143:
7108:
7088:
7068:
7048:
7028:
7002:
6973:
6932:
6900:
6880:
6865:. Consider the set
6860:
6819:
6784:
6746:
6726:
6706:
6686:
6666:
6637:
6602:
6582:
6547:
6521:
6489:
6469:
6434:
6414:
6382:
6350:
6315:
6283:
6251:
6175:
6046:
5939:
5841:
5839:{\displaystyle x\in }
5771:
5709:
5541:non-standard calculus
5295:attains its supremum
5275:)} that converges to
5237:
5183:
5014:Dedekind-completeness
4987:
4955:
4935:
4897:
4865:
4845:
4811:
4791:
4771:
4751:
4731:
4693:
4673:
4613:
4592:, hence there exists
4587:
4567:
4547:
4518:
4498:
4460:
4440:
4396:
4364:
4344:
4310:
4290:
4270:
4250:
4230:
4186:
4166:
4106:
4085:, hence there exists
4080:
4060:
4040:
4011:
3985:
3965:
3945:
3913:
3893:
3873:
3850:
3830:
3815:So far, we know that
3807:
3787:
3749:
3714:
3679:
3659:
3621:
3601:
3541:
3520:, hence there exists
3515:
3495:
3472:
3452:
3432:
3412:
3392:
3372:
3346:
3317:
3285:
3256:
3236:
3204:
3175:
3143:
3123:
3092:
3060:
3039:
3015:
2975:, which implies that
2970:
2950:
2886:
2862:
2842:
2813:
2768:
2748:
2728:
2708:
2676:
2656:
2621:
2562:
2530:
2479:
2437:
2392:
2372:
2340:
2325:Suppose the function
2313:
2278:
2243:
2184:
2142:
2122:
2099:
2059:
1999:
1939:
1907:
1887:
1853:
1823:
1795:
1775:
1755:
1726:
1694:
1659:
1633:
1601:
1599:{\displaystyle V,\ W}
1562:
1517:
1491:
1459:
1457:{\displaystyle V,\ W}
1419:
1399:
1350:
1338:
1278:
1242:
1212:
1180:
1150:
1118:
1079:
1059:
1057:{\displaystyle (0,1]}
1027:
984:
982:{\displaystyle (0,1]}
952:
908:
888:
856:
804:
772:
682:
647:
627:
607:
585:
516:
496:
472:
440:
414:
327:
295:
275:
247:
227:
183:
163:extreme value theorem
148:
116:
91:
11317:Theorems in calculus
11224:Miscellaneous topics
11164:hyperbolic functions
11149:irrational functions
11027:Exponential function
10880:Sequences and series
10646:Integration by parts
10049:Historia Mathematica
9914:
9860:
9780:of the subsequence {
9749:
9707:
9588:
9534:
9486:
9454:
9434:
9386:
9359:
9331:
9304:
9256:
9224:
9204:
9150:
9109:
9089:
9063:
9043:
9023:
9003:
8965:
8931:
8911:
8873:
8853:
8785:
8759:
8739:
8719:
8692:
8669:
8637:
8590:
8552:
8532:
8484:
8457:
8429:
8402:
8354:
8322:
8302:
8248:
8207:
8187:
8161:
8141:
8121:
8101:
8057:
8023:
8003:
7959:
7939:
7871:
7845:
7825:
7805:
7778:
7733:
7698:
7660:
7640:
7620:
7594:
7574:
7542:
7522:
7518:is bounded above by
7502:
7479:
7441:
7407:
7387:
7349:
7329:
7261:
7235:
7215:
7195:
7172:
7152:
7117:
7097:
7077:
7057:
7037:
7033:is another point in
7011:
7007: ; moreover if
6985:
6941:
6909:
6889:
6869:
6828:
6793:
6758:
6735:
6715:
6695:
6675:
6664:{\displaystyle f(a)}
6646:
6611:
6591:
6556:
6530:
6498:
6478:
6443:
6423:
6391:
6359:
6355:is bounded above on
6348:{\displaystyle f(x)}
6330:
6292:
6260:
6249:{\displaystyle f(x)}
6231:
6062:
6051:. By continuity of
5948:
5869:
5792:
5726:
5639:
5212:
5192:and, as is closed,
5158:
4964:
4944:
4906:
4874:
4854:
4820:
4800:
4780:
4760:
4740:
4702:
4682:
4622:
4596:
4576:
4556:
4530:
4507:
4469:
4449:
4405:
4373:
4353:
4319:
4299:
4279:
4259:
4239:
4195:
4175:
4115:
4089:
4069:
4049:
4023:
3994:
3974:
3954:
3922:
3902:
3882:
3878:is bounded above by
3862:
3839:
3819:
3796:
3758:
3723:
3688:
3668:
3630:
3610:
3550:
3524:
3504:
3484:
3461:
3441:
3421:
3401:
3381:
3355:
3344:{\displaystyle f(a)}
3326:
3294:
3283:{\displaystyle f(x)}
3265:
3245:
3213:
3202:{\displaystyle f(x)}
3184:
3152:
3132:
3112:
3069:
3065:is bounded above on
3049:
3025:
2979:
2959:
2895:
2875:
2851:
2840:{\displaystyle f(x)}
2822:
2777:
2757:
2737:
2717:
2685:
2665:
2630:
2575:
2539:
2491:
2446:
2401:
2381:
2349:
2329:
2287:
2252:
2241:{\displaystyle f(x)}
2223:
2173:
2131:
2108:
2088:
2076:Proving the theorems
2008:
1948:
1916:
1896:
1862:
1842:
1812:
1784:
1764:
1753:{\displaystyle f(K)}
1735:
1707:
1668:
1642:
1610:
1581:
1526:
1500:
1468:
1439:
1430:Heine–Borel property
1408:
1347:
1287:
1251:
1224:
1201:
1167:
1127:
1092:
1068:
1036:
995:
961:
919:
897:
865:
815:
781:
746:
656:
636:
616:
596:
525:
505:
485:
449:
429:
336:
304:
284:
264:
236:
204:
172:
125:
114:{\displaystyle f(x)}
96:
11211:List of derivatives
11047:History of calculus
10962:Cauchy condensation
10859:Exterior derivative
10816:Lagrange multiplier
10552:Maximum and minimum
10383:Limit of a sequence
10371:Limit of a function
10318:Graph of a function
10298:Continuous function
10005: —
9679: —
9348:{\displaystyle d/2}
8707:{\displaystyle s=b}
8446:{\displaystyle d/2}
6494:to the subinterval
6208:to be a maximum of
5624:such that 0 ≤
5475:, which means that
5423:is continuous on .
4545:{\displaystyle s=b}
3990:we can deduce that
2217: —
2214:Boundedness Theorem
2084:and the maximum of
1836: —
1575: —
1426:Heine–Borel theorem
423:boundedness theorem
11144:rational functions
11111:Method of Fluxions
10957:Alternating series
10854:Differential forms
10836:Partial derivative
10796:Divergence theorem
10678:Quadratic integral
10446:Leibniz's notation
10436:Mean value theorem
10421:Partial derivative
10366:Indeterminate form
10215:Weisstein, Eric W.
10003:
9988:
9932:
9905:
9758:
9731:
9701:
9677:
9662:
9606:
9579:
9492:
9472:
9440:
9420:
9372:
9345:
9317:
9290:
9242:
9210:
9190:
9136:
9095:
9075:
9049:
9039:say, belonging to
9029:
9009:
8989:
8951:
8917:
8897:
8859:
8839:
8771:
8745:
8725:
8715:. As
8704:
8675:
8655:
8623:
8576:
8538:
8518:
8470:
8443:
8415:
8388:
8340:
8308:
8288:
8234:
8193:
8173:
8147:
8137:say, belonging to
8127:
8107:
8087:
8043:
8009:
7989:
7945:
7925:
7857:
7831:
7821:is continuous at
7811:
7790:
7760:
7719:
7681:
7646:
7626:
7606:
7580:
7570:: let us call it
7560:
7528:
7508:
7485:
7465:
7427:
7393:
7373:
7335:
7315:
7247:
7221:
7201:
7178:
7158:
7138:
7103:
7083:
7063:
7043:
7023:
6997:
6968:
6927:
6895:
6875:
6855:
6814:
6779:
6741:
6721:
6701:
6681:
6661:
6632:
6597:
6577:
6542:
6516:
6484:
6464:
6429:
6419:. Let us call it
6409:
6377:
6345:
6324:
6310:
6278:
6246:
6170:
6041:
5934:
5865:to the inequality
5836:
5766:
5704:
5615:transfer principle
5539:In the setting of
5537:
5312:
5232:
5178:
5006:
4982:
4950:
4930:
4892:
4860:
4840:
4806:
4786:
4766:
4746:
4726:
4688:
4668:
4608:
4582:
4562:
4542:
4513:
4493:
4455:
4435:
4391:
4359:
4339:
4305:
4285:
4265:
4245:
4225:
4181:
4161:
4101:
4075:
4055:
4035:
4006:
3980:
3960:
3940:
3918:has a supremum in
3908:
3888:
3868:
3845:
3825:
3802:
3782:
3744:
3709:
3674:
3654:
3616:
3596:
3536:
3510:
3490:
3467:
3447:
3437:. In other words
3427:
3407:
3387:
3367:
3341:
3312:
3280:
3251:
3231:
3199:
3170:
3138:
3118:
3106:
3087:
3055:
3034:
3010:
2965:
2945:
2881:
2857:
2837:
2808:
2763:
2743:
2723:
2703:
2671:
2651:
2616:
2557:
2525:
2474:
2432:
2397:, there exists an
2387:
2367:
2335:
2323:
2308:
2273:
2238:
2215:
2179:
2137:
2120:{\displaystyle -f}
2117:
2094:
2054:
2041:
1994:
1981:
1934:
1902:
1882:
1848:
1834:
1818:
1790:
1770:
1750:
1721:
1703:In particular, if
1689:
1654:
1628:
1596:
1573:
1557:
1512:
1486:
1454:
1414:
1394:
1333:
1273:
1237:
1207:
1191:topological spaces
1175:
1145:
1113:
1074:
1054:
1022:
979:
947:
903:
883:
851:
799:
767:
677:
642:
622:
602:
580:
511:
491:
467:
435:
409:
322:
290:
270:
242:
222:
178:
155:
143:
111:
11304:
11303:
11230:Complex calculus
11219:
11218:
11100:Law of Continuity
11032:Natural logarithm
11017:Bernoulli numbers
11008:Special functions
10967:Direct comparison
10831:Multiple integral
10705:Integral equation
10601:Integral calculus
10532:Stationary points
10506:Other techniques
10451:Newton's notation
10416:Second derivative
10308:Finite difference
10001:
9917:
9699:
9675:
9591:
9495:{\displaystyle M}
9443:{\displaystyle x}
9213:{\displaystyle x}
9098:{\displaystyle L}
9052:{\displaystyle L}
9032:{\displaystyle e}
9012:{\displaystyle s}
8920:{\displaystyle f}
8862:{\displaystyle x}
8748:{\displaystyle s}
8728:{\displaystyle f}
8678:{\displaystyle s}
8541:{\displaystyle x}
8311:{\displaystyle x}
8196:{\displaystyle L}
8150:{\displaystyle L}
8130:{\displaystyle e}
8110:{\displaystyle s}
8012:{\displaystyle f}
7948:{\displaystyle x}
7834:{\displaystyle s}
7814:{\displaystyle f}
7801:. As
7649:{\displaystyle f}
7629:{\displaystyle s}
7583:{\displaystyle s}
7531:{\displaystyle b}
7511:{\displaystyle L}
7488:{\displaystyle L}
7396:{\displaystyle f}
7338:{\displaystyle x}
7224:{\displaystyle a}
7204:{\displaystyle f}
7181:{\displaystyle a}
7161:{\displaystyle L}
7141:{\displaystyle M}
7106:{\displaystyle L}
7086:{\displaystyle e}
7066:{\displaystyle a}
7046:{\displaystyle L}
6898:{\displaystyle x}
6878:{\displaystyle L}
6744:{\displaystyle b}
6724:{\displaystyle a}
6704:{\displaystyle x}
6684:{\displaystyle M}
6635:{\displaystyle M}
6600:{\displaystyle M}
6580:{\displaystyle M}
6487:{\displaystyle f}
6467:{\displaystyle M}
6432:{\displaystyle M}
6322:
6256:is continuous on
5846:, so that
5582:"runs" from 0 to
5535:
5407:on . Therefore,
5344:least upper bound
5310:
5200:is continuous at
5004:
4953:{\displaystyle f}
4863:{\displaystyle f}
4809:{\displaystyle e}
4789:{\displaystyle B}
4769:{\displaystyle s}
4749:{\displaystyle f}
4691:{\displaystyle x}
4585:{\displaystyle s}
4565:{\displaystyle f}
4516:{\displaystyle s}
4458:{\displaystyle f}
4362:{\displaystyle f}
4308:{\displaystyle e}
4288:{\displaystyle B}
4268:{\displaystyle s}
4248:{\displaystyle f}
4184:{\displaystyle x}
4078:{\displaystyle s}
4065:is continuous at
4058:{\displaystyle f}
3983:{\displaystyle B}
3963:{\displaystyle s}
3911:{\displaystyle B}
3898:. Hence the set
3891:{\displaystyle b}
3871:{\displaystyle B}
3848:{\displaystyle a}
3828:{\displaystyle B}
3805:{\displaystyle B}
3677:{\displaystyle f}
3619:{\displaystyle x}
3513:{\displaystyle a}
3493:{\displaystyle f}
3470:{\displaystyle a}
3450:{\displaystyle B}
3430:{\displaystyle B}
3410:{\displaystyle e}
3390:{\displaystyle a}
3254:{\displaystyle a}
3141:{\displaystyle p}
3121:{\displaystyle B}
3108:Consider the set
3105:Alternative proof
3104:
3058:{\displaystyle f}
2968:{\displaystyle k}
2884:{\displaystyle x}
2860:{\displaystyle f}
2766:{\displaystyle x}
2753:is continuous at
2746:{\displaystyle f}
2726:{\displaystyle x}
2674:{\displaystyle x}
2484:. This defines a
2390:{\displaystyle n}
2338:{\displaystyle f}
2321:
2311:{\displaystyle .}
2276:{\displaystyle ,}
2248:is continuous on
2213:
2182:{\displaystyle f}
2165:converges to the
2140:{\displaystyle f}
2097:{\displaystyle f}
2026:
1966:
1905:{\displaystyle f}
1851:{\displaystyle K}
1832:
1821:{\displaystyle K}
1793:{\displaystyle f}
1773:{\displaystyle K}
1699:is also compact.
1664:is compact, then
1592:
1571:
1450:
1417:{\displaystyle K}
1210:{\displaystyle K}
1077:{\displaystyle 1}
945:
906:{\displaystyle 1}
849:
680:{\displaystyle ,}
645:{\displaystyle f}
625:{\displaystyle m}
605:{\displaystyle M}
514:{\displaystyle M}
494:{\displaystyle m}
438:{\displaystyle f}
293:{\displaystyle d}
273:{\displaystyle c}
245:{\displaystyle f}
181:{\displaystyle f}
86:
85:
78:
11329:
11234:Contour integral
11132:
11131:
10982:Limit comparison
10891:Types of series
10850:Advanced topics
10841:Surface integral
10685:Trapezoidal rule
10624:Basic properties
10619:Riemann integral
10567:Taylor's theorem
10293:Concave function
10288:Binomial theorem
10265:
10258:
10251:
10242:
10241:
10228:
10227:
10183:
10153:
10125:
10124:
10112:
10101:
10095:
10094:
10074:
10065:
10064:
10044:
10016:
10015:
10006:
9997:
9995:
9994:
9989:
9931:
9912:
9911:
9906:
9767:
9765:
9764:
9759:
9740:
9738:
9737:
9732:
9690:
9689:
9680:
9671:
9669:
9668:
9663:
9605:
9586:
9585:
9580:
9501:
9499:
9498:
9493:
9481:
9479:
9478:
9475:{\displaystyle }
9473:
9449:
9447:
9446:
9441:
9429:
9427:
9426:
9421:
9419:
9418:
9381:
9379:
9378:
9373:
9371:
9370:
9354:
9352:
9351:
9346:
9341:
9326:
9324:
9323:
9318:
9316:
9315:
9299:
9297:
9296:
9291:
9289:
9288:
9251:
9249:
9248:
9245:{\displaystyle }
9243:
9219:
9217:
9216:
9211:
9199:
9197:
9196:
9191:
9162:
9161:
9145:
9143:
9142:
9137:
9104:
9102:
9101:
9096:
9084:
9082:
9081:
9076:
9058:
9056:
9055:
9050:
9038:
9036:
9035:
9030:
9018:
9016:
9015:
9010:
8998:
8996:
8995:
8992:{\displaystyle }
8990:
8961:on the interval
8960:
8958:
8957:
8952:
8947:
8926:
8924:
8923:
8918:
8906:
8904:
8903:
8900:{\displaystyle }
8898:
8868:
8866:
8865:
8860:
8848:
8846:
8845:
8840:
8835:
8824:
8792:
8780:
8778:
8777:
8772:
8754:
8752:
8751:
8746:
8734:
8732:
8731:
8726:
8713:
8711:
8710:
8705:
8684:
8682:
8681:
8676:
8664:
8662:
8661:
8656:
8632:
8630:
8629:
8624:
8585:
8583:
8582:
8579:{\displaystyle }
8577:
8547:
8545:
8544:
8539:
8527:
8525:
8524:
8519:
8517:
8516:
8479:
8477:
8476:
8471:
8469:
8468:
8452:
8450:
8449:
8444:
8439:
8424:
8422:
8421:
8416:
8414:
8413:
8397:
8395:
8394:
8389:
8387:
8386:
8349:
8347:
8346:
8343:{\displaystyle }
8341:
8317:
8315:
8314:
8309:
8297:
8295:
8294:
8289:
8260:
8259:
8243:
8241:
8240:
8235:
8202:
8200:
8199:
8194:
8182:
8180:
8179:
8174:
8156:
8154:
8153:
8148:
8136:
8134:
8133:
8128:
8116:
8114:
8113:
8108:
8096:
8094:
8093:
8090:{\displaystyle }
8088:
8053:on the interval
8052:
8050:
8049:
8044:
8039:
8018:
8016:
8015:
8010:
7998:
7996:
7995:
7992:{\displaystyle }
7990:
7954:
7952:
7951:
7946:
7934:
7932:
7931:
7926:
7921:
7910:
7878:
7866:
7864:
7863:
7858:
7840:
7838:
7837:
7832:
7820:
7818:
7817:
7812:
7799:
7797:
7796:
7791:
7769:
7767:
7766:
7761:
7728:
7726:
7725:
7720:
7690:
7688:
7687:
7682:
7655:
7653:
7652:
7647:
7635:
7633:
7632:
7627:
7615:
7613:
7612:
7607:
7589:
7587:
7586:
7581:
7569:
7567:
7566:
7563:{\displaystyle }
7561:
7537:
7535:
7534:
7529:
7517:
7515:
7514:
7509:
7494:
7492:
7491:
7486:
7474:
7472:
7471:
7468:{\displaystyle }
7466:
7437:on the interval
7436:
7434:
7433:
7428:
7423:
7402:
7400:
7399:
7394:
7382:
7380:
7379:
7376:{\displaystyle }
7374:
7344:
7342:
7341:
7336:
7324:
7322:
7321:
7316:
7311:
7300:
7268:
7256:
7254:
7253:
7248:
7230:
7228:
7227:
7222:
7210:
7208:
7207:
7202:
7187:
7185:
7184:
7179:
7167:
7165:
7164:
7159:
7147:
7145:
7144:
7139:
7112:
7110:
7109:
7104:
7092:
7090:
7089:
7084:
7072:
7070:
7069:
7064:
7052:
7050:
7049:
7044:
7032:
7030:
7029:
7024:
7006:
7004:
7003:
6998:
6977:
6975:
6974:
6969:
6936:
6934:
6933:
6930:{\displaystyle }
6928:
6904:
6902:
6901:
6896:
6884:
6882:
6881:
6876:
6864:
6862:
6861:
6856:
6823:
6821:
6820:
6815:
6788:
6786:
6785:
6780:
6750:
6748:
6747:
6742:
6730:
6728:
6727:
6722:
6710:
6708:
6707:
6702:
6690:
6688:
6687:
6682:
6670:
6668:
6667:
6662:
6641:
6639:
6638:
6633:
6606:
6604:
6603:
6598:
6586:
6584:
6583:
6578:
6551:
6549:
6548:
6543:
6525:
6523:
6522:
6519:{\displaystyle }
6517:
6493:
6491:
6490:
6485:
6473:
6471:
6470:
6465:
6438:
6436:
6435:
6430:
6418:
6416:
6415:
6412:{\displaystyle }
6410:
6386:
6384:
6383:
6380:{\displaystyle }
6378:
6354:
6352:
6351:
6346:
6319:
6317:
6316:
6313:{\displaystyle }
6311:
6287:
6285:
6284:
6281:{\displaystyle }
6279:
6255:
6253:
6252:
6247:
6200:), for all real
6179:
6177:
6176:
6171:
6148:
6147:
6146:
6145:
6128:
6105:
6104:
6103:
6102:
6085:
6084:
6072:
6055: we have
6050:
6048:
6047:
6042:
6034:
6033:
6021:
6020:
6008:
5991:
5990:
5989:
5988:
5971:
5970:
5958:
5943:
5941:
5940:
5935:
5930:
5929:
5917:
5916:
5901:
5900:
5899:
5898:
5881:
5880:
5845:
5843:
5842:
5837:
5832:
5831:
5813:
5812:
5775:
5773:
5772:
5767:
5762:
5761:
5760:
5759:
5742:
5714: for all
5713:
5711:
5710:
5705:
5700:
5699:
5687:
5686:
5671:
5670:
5669:
5668:
5651:
5650:
5586:. The function
5506:
5490:
5474:
5451:
5422:
5406:
5376:
5364:
5341:
5242:)} converges to
5241:
5239:
5238:
5233:
5231:
5230:
5229:
5228:
5204:, the sequence {
5187:
5185:
5184:
5179:
5177:
5176:
5175:
5174:
5143:)} converges to
4991:
4989:
4988:
4985:{\displaystyle }
4983:
4959:
4957:
4956:
4951:
4939:
4937:
4936:
4933:{\displaystyle }
4931:
4901:
4899:
4898:
4895:{\displaystyle }
4893:
4869:
4867:
4866:
4861:
4849:
4847:
4846:
4841:
4836:
4815:
4813:
4812:
4807:
4795:
4793:
4792:
4787:
4775:
4773:
4772:
4767:
4755:
4753:
4752:
4747:
4735:
4733:
4732:
4729:{\displaystyle }
4727:
4697:
4695:
4694:
4689:
4677:
4675:
4674:
4669:
4661:
4629:
4617:
4615:
4614:
4609:
4591:
4589:
4588:
4583:
4571:
4569:
4568:
4563:
4551:
4549:
4548:
4543:
4522:
4520:
4519:
4514:
4502:
4500:
4499:
4496:{\displaystyle }
4494:
4464:
4462:
4461:
4456:
4444:
4442:
4441:
4438:{\displaystyle }
4436:
4400:
4398:
4397:
4394:{\displaystyle }
4392:
4368:
4366:
4365:
4360:
4348:
4346:
4345:
4340:
4335:
4314:
4312:
4311:
4306:
4294:
4292:
4291:
4286:
4274:
4272:
4271:
4266:
4254:
4252:
4251:
4246:
4234:
4232:
4231:
4228:{\displaystyle }
4226:
4190:
4188:
4187:
4182:
4170:
4168:
4167:
4162:
4154:
4122:
4110:
4108:
4107:
4102:
4084:
4082:
4081:
4076:
4064:
4062:
4061:
4056:
4044:
4042:
4041:
4036:
4015:
4013:
4012:
4007:
3989:
3987:
3986:
3981:
3969:
3967:
3966:
3961:
3949:
3947:
3946:
3943:{\displaystyle }
3941:
3917:
3915:
3914:
3909:
3897:
3895:
3894:
3889:
3877:
3875:
3874:
3869:
3854:
3852:
3851:
3846:
3834:
3832:
3831:
3826:
3811:
3809:
3808:
3803:
3791:
3789:
3788:
3785:{\displaystyle }
3783:
3754:on the interval
3753:
3751:
3750:
3745:
3718:
3716:
3715:
3710:
3683:
3681:
3680:
3675:
3663:
3661:
3660:
3657:{\displaystyle }
3655:
3625:
3623:
3622:
3617:
3605:
3603:
3602:
3597:
3589:
3557:
3545:
3543:
3542:
3537:
3519:
3517:
3516:
3511:
3499:
3497:
3496:
3491:
3476:
3474:
3473:
3468:
3456:
3454:
3453:
3448:
3436:
3434:
3433:
3428:
3416:
3414:
3413:
3408:
3396:
3394:
3393:
3388:
3376:
3374:
3373:
3368:
3350:
3348:
3347:
3342:
3321:
3319:
3318:
3315:{\displaystyle }
3313:
3289:
3287:
3286:
3281:
3260:
3258:
3257:
3252:
3241:. We note that
3240:
3238:
3237:
3234:{\displaystyle }
3232:
3208:
3206:
3205:
3200:
3179:
3177:
3176:
3173:{\displaystyle }
3171:
3147:
3145:
3144:
3139:
3127:
3125:
3124:
3119:
3096:
3094:
3093:
3090:{\displaystyle }
3088:
3064:
3062:
3061:
3056:
3043:
3041:
3040:
3035:
3019:
3017:
3016:
3011:
3006:
3005:
3004:
3003:
2998:
2974:
2972:
2971:
2966:
2954:
2952:
2951:
2946:
2938:
2937:
2922:
2921:
2920:
2919:
2914:
2890:
2888:
2887:
2882:
2866:
2864:
2863:
2858:
2846:
2844:
2843:
2838:
2817:
2815:
2814:
2809:
2804:
2803:
2802:
2801:
2796:
2772:
2770:
2769:
2764:
2752:
2750:
2749:
2744:
2732:
2730:
2729:
2724:
2712:
2710:
2709:
2706:{\displaystyle }
2704:
2680:
2678:
2677:
2672:
2660:
2658:
2657:
2652:
2647:
2646:
2645:
2625:
2623:
2622:
2617:
2615:
2614:
2613:
2597:
2596:
2595:
2594:
2567:is bounded, the
2566:
2564:
2563:
2560:{\displaystyle }
2558:
2534:
2532:
2531:
2526:
2524:
2523:
2522:
2506:
2505:
2483:
2481:
2480:
2475:
2464:
2463:
2441:
2439:
2438:
2433:
2413:
2412:
2396:
2394:
2393:
2388:
2376:
2374:
2373:
2370:{\displaystyle }
2368:
2344:
2342:
2341:
2336:
2317:
2315:
2314:
2309:
2282:
2280:
2279:
2274:
2247:
2245:
2244:
2239:
2218:
2188:
2186:
2185:
2180:
2146:
2144:
2143:
2138:
2126:
2124:
2123:
2118:
2103:
2101:
2100:
2095:
2063:
2061:
2060:
2055:
2040:
2003:
2001:
2000:
1995:
1980:
1943:
1941:
1940:
1935:
1911:
1909:
1908:
1903:
1891:
1889:
1888:
1883:
1881:
1857:
1855:
1854:
1849:
1837:
1827:
1825:
1824:
1819:
1799:
1797:
1796:
1791:
1779:
1777:
1776:
1771:
1759:
1757:
1756:
1751:
1730:
1728:
1727:
1722:
1720:
1698:
1696:
1695:
1690:
1663:
1661:
1660:
1655:
1637:
1635:
1634:
1629:
1605:
1603:
1602:
1597:
1590:
1576:
1566:
1564:
1563:
1558:
1541:
1540:
1521:
1519:
1518:
1513:
1495:
1493:
1492:
1487:
1463:
1461:
1460:
1455:
1448:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1387:
1386:
1385:
1384:
1369:
1364:
1342:
1340:
1339:
1334:
1332:
1331:
1330:
1329:
1306:
1305:
1304:
1303:
1282:
1280:
1279:
1274:
1266:
1265:
1246:
1244:
1243:
1238:
1236:
1235:
1216:
1214:
1213:
1208:
1184:
1182:
1181:
1176:
1174:
1154:
1152:
1151:
1148:{\displaystyle }
1146:
1122:
1120:
1119:
1114:
1083:
1081:
1080:
1075:
1063:
1061:
1060:
1055:
1031:
1029:
1028:
1023:
988:
986:
985:
980:
956:
954:
953:
948:
946:
938:
912:
910:
909:
904:
892:
890:
889:
884:
860:
858:
857:
852:
850:
848:
834:
808:
806:
805:
800:
776:
774:
773:
768:
696:Karl Weierstrass
686:
684:
683:
678:
652:on the interval
651:
649:
648:
643:
631:
629:
628:
623:
611:
609:
608:
603:
589:
587:
586:
581:
520:
518:
517:
512:
500:
498:
497:
492:
476:
474:
473:
470:{\displaystyle }
468:
444:
442:
441:
436:
418:
416:
415:
410:
331:
329:
328:
325:{\displaystyle }
323:
299:
297:
296:
291:
279:
277:
276:
271:
251:
249:
248:
243:
231:
229:
228:
225:{\displaystyle }
223:
187:
185:
184:
179:
152:
150:
149:
146:{\displaystyle }
144:
120:
118:
117:
112:
81:
74:
70:
67:
61:
56:this article by
47:inline citations
34:
33:
26:
11337:
11336:
11332:
11331:
11330:
11328:
11327:
11326:
11307:
11306:
11305:
11300:
11296:Steinmetz solid
11281:Integration Bee
11215:
11197:
11123:
11065:Colin Maclaurin
11041:
11009:
11003:
10875:
10869:Tensor calculus
10846:Volume integral
10782:
10757:Basic theorems
10720:Vector calculus
10714:
10595:
10562:Newton's method
10397:
10376:One-sided limit
10352:
10333:Rolle's theorem
10323:Linear function
10274:
10269:
10190:
10180:
10150:
10134:
10132:Further reading
10129:
10128:
10121:
10110:
10102:
10098:
10091:
10075:
10068:
10045:
10041:
10036:
10027:
10013:
10008:
10004:
9921:
9915:
9861:
9858:
9857:
9851:
9849:
9822:
9821:
9792:
9791:
9750:
9747:
9746:
9708:
9705:
9704:
9697:
9687:
9682:
9678:
9595:
9589:
9535:
9532:
9531:
9521:semi-continuity
9519:is weakened to
9513:
9508:
9487:
9484:
9483:
9455:
9452:
9451:
9435:
9432:
9431:
9414:
9410:
9387:
9384:
9383:
9366:
9362:
9360:
9357:
9356:
9337:
9332:
9329:
9328:
9311:
9307:
9305:
9302:
9301:
9284:
9280:
9257:
9254:
9253:
9225:
9222:
9221:
9205:
9202:
9201:
9157:
9153:
9151:
9148:
9147:
9110:
9107:
9106:
9090:
9087:
9086:
9064:
9061:
9060:
9044:
9041:
9040:
9024:
9021:
9020:
9004:
9001:
9000:
8966:
8963:
8962:
8943:
8932:
8929:
8928:
8912:
8909:
8908:
8874:
8871:
8870:
8854:
8851:
8850:
8831:
8820:
8788:
8786:
8783:
8782:
8760:
8757:
8756:
8755:, there exists
8740:
8737:
8736:
8720:
8717:
8716:
8693:
8690:
8689:
8670:
8667:
8666:
8638:
8635:
8634:
8591:
8588:
8587:
8553:
8550:
8549:
8533:
8530:
8529:
8512:
8508:
8485:
8482:
8481:
8464:
8460:
8458:
8455:
8454:
8435:
8430:
8427:
8426:
8409:
8405:
8403:
8400:
8399:
8382:
8378:
8355:
8352:
8351:
8323:
8320:
8319:
8303:
8300:
8299:
8255:
8251:
8249:
8246:
8245:
8208:
8205:
8204:
8188:
8185:
8184:
8162:
8159:
8158:
8142:
8139:
8138:
8122:
8119:
8118:
8102:
8099:
8098:
8058:
8055:
8054:
8035:
8024:
8021:
8020:
8004:
8001:
8000:
7960:
7957:
7956:
7940:
7937:
7936:
7917:
7906:
7874:
7872:
7869:
7868:
7846:
7843:
7842:
7841:, there exists
7826:
7823:
7822:
7806:
7803:
7802:
7779:
7776:
7775:
7734:
7731:
7730:
7699:
7696:
7695:
7661:
7658:
7657:
7641:
7638:
7637:
7621:
7618:
7617:
7595:
7592:
7591:
7575:
7572:
7571:
7543:
7540:
7539:
7523:
7520:
7519:
7503:
7500:
7499:
7480:
7477:
7476:
7442:
7439:
7438:
7419:
7408:
7405:
7404:
7388:
7385:
7384:
7350:
7347:
7346:
7330:
7327:
7326:
7307:
7296:
7264:
7262:
7259:
7258:
7236:
7233:
7232:
7216:
7213:
7212:
7196:
7193:
7192:
7173:
7170:
7169:
7153:
7150:
7149:
7118:
7115:
7114:
7098:
7095:
7094:
7093:also belong to
7078:
7075:
7074:
7058:
7055:
7054:
7038:
7035:
7034:
7012:
7009:
7008:
6986:
6983:
6982:
6942:
6939:
6938:
6910:
6907:
6906:
6890:
6887:
6886:
6870:
6867:
6866:
6829:
6826:
6825:
6794:
6791:
6790:
6759:
6756:
6755:
6736:
6733:
6732:
6716:
6713:
6712:
6711:increases from
6696:
6693:
6692:
6676:
6673:
6672:
6647:
6644:
6643:
6642:increases from
6612:
6609:
6608:
6592:
6589:
6588:
6557:
6554:
6553:
6552:has a supremum
6531:
6528:
6527:
6499:
6496:
6495:
6479:
6476:
6475:
6444:
6441:
6440:
6424:
6421:
6420:
6392:
6389:
6388:
6360:
6357:
6356:
6331:
6328:
6327:
6293:
6290:
6289:
6261:
6258:
6257:
6232:
6229:
6228:
6222:
6217:
6141:
6137:
6136:
6132:
6121:
6098:
6094:
6093:
6089:
6080:
6076:
6065:
6063:
6060:
6059:
6029:
6025:
6016:
6012:
6001:
5984:
5980:
5979:
5975:
5966:
5962:
5951:
5949:
5946:
5945:
5925:
5921:
5912:
5908:
5894:
5890:
5889:
5885:
5876:
5872:
5870:
5867:
5866:
5855:
5821:
5817:
5808:
5804:
5793:
5790:
5789:
5755:
5751:
5750:
5746:
5735:
5727:
5724:
5723:
5695:
5691:
5682:
5678:
5664:
5660:
5659:
5655:
5646:
5642:
5640:
5637:
5636:
5630:
5623:
5611:
5568:
5533:
5528:
5492:
5476:
5457:
5435:
5408:
5394:
5366:
5351:
5315:
5308:
5273:
5262:
5261:
5224:
5220:
5219:
5215:
5213:
5210:
5209:
5170:
5166:
5165:
5161:
5159:
5156:
5155:
5141:
5122:
5095:
5088:
5069:
5002:
4997:
4992:.
4965:
4962:
4961:
4945:
4942:
4941:
4907:
4904:
4903:
4902:which overlaps
4875:
4872:
4871:
4855:
4852:
4851:
4832:
4821:
4818:
4817:
4801:
4798:
4797:
4781:
4778:
4777:
4761:
4758:
4757:
4741:
4738:
4737:
4703:
4700:
4699:
4683:
4680:
4679:
4657:
4625:
4623:
4620:
4619:
4597:
4594:
4593:
4577:
4574:
4573:
4557:
4554:
4553:
4531:
4528:
4527:
4508:
4505:
4504:
4470:
4467:
4466:
4450:
4447:
4446:
4406:
4403:
4402:
4401:which overlaps
4374:
4371:
4370:
4354:
4351:
4350:
4331:
4320:
4317:
4316:
4300:
4297:
4296:
4280:
4277:
4276:
4260:
4257:
4256:
4240:
4237:
4236:
4196:
4193:
4192:
4176:
4173:
4172:
4150:
4118:
4116:
4113:
4112:
4090:
4087:
4086:
4070:
4067:
4066:
4050:
4047:
4046:
4024:
4021:
4020:
3995:
3992:
3991:
3975:
3972:
3971:
3955:
3952:
3951:
3923:
3920:
3919:
3903:
3900:
3899:
3883:
3880:
3879:
3863:
3860:
3859:
3840:
3837:
3836:
3820:
3817:
3816:
3797:
3794:
3793:
3759:
3756:
3755:
3724:
3721:
3720:
3689:
3686:
3685:
3669:
3666:
3665:
3631:
3628:
3627:
3611:
3608:
3607:
3585:
3553:
3551:
3548:
3547:
3525:
3522:
3521:
3505:
3502:
3501:
3485:
3482:
3481:
3462:
3459:
3458:
3442:
3439:
3438:
3422:
3419:
3418:
3417:also belong to
3402:
3399:
3398:
3382:
3379:
3378:
3356:
3353:
3352:
3327:
3324:
3323:
3295:
3292:
3291:
3266:
3263:
3262:
3246:
3243:
3242:
3214:
3211:
3210:
3185:
3182:
3181:
3153:
3150:
3149:
3133:
3130:
3129:
3113:
3110:
3109:
3102:
3070:
3067:
3066:
3050:
3047:
3046:
3026:
3023:
3022:
2999:
2994:
2993:
2992:
2988:
2980:
2977:
2976:
2960:
2957:
2956:
2933:
2929:
2915:
2910:
2909:
2908:
2904:
2896:
2893:
2892:
2876:
2873:
2872:
2852:
2849:
2848:
2823:
2820:
2819:
2797:
2792:
2791:
2790:
2786:
2778:
2775:
2774:
2773:, we know that
2758:
2755:
2754:
2738:
2735:
2734:
2718:
2715:
2714:
2686:
2683:
2682:
2666:
2663:
2662:
2641:
2637:
2636:
2631:
2628:
2627:
2609:
2602:
2598:
2590:
2586:
2585:
2581:
2576:
2573:
2572:
2540:
2537:
2536:
2518:
2511:
2507:
2501:
2497:
2492:
2489:
2488:
2459:
2455:
2447:
2444:
2443:
2408:
2404:
2402:
2399:
2398:
2382:
2379:
2378:
2350:
2347:
2346:
2330:
2327:
2326:
2319:
2288:
2285:
2284:
2253:
2250:
2249:
2224:
2221:
2220:
2216:
2210:
2174:
2171:
2170:
2132:
2129:
2128:
2109:
2106:
2105:
2089:
2086:
2085:
2078:
2066:
2030:
2009:
2006:
2005:
1970:
1949:
1946:
1945:
1917:
1914:
1913:
1897:
1894:
1893:
1877:
1863:
1860:
1859:
1843:
1840:
1839:
1835:
1813:
1810:
1809:
1785:
1782:
1781:
1765:
1762:
1761:
1736:
1733:
1732:
1716:
1708:
1705:
1704:
1701:
1669:
1666:
1665:
1643:
1640:
1639:
1611:
1608:
1607:
1582:
1579:
1578:
1574:
1533:
1529:
1527:
1524:
1523:
1501:
1498:
1497:
1469:
1466:
1465:
1440:
1437:
1436:
1409:
1406:
1405:
1380:
1376:
1375:
1371:
1365:
1354:
1348:
1345:
1344:
1325:
1321:
1320:
1316:
1299:
1295:
1294:
1290:
1288:
1285:
1284:
1261:
1257:
1252:
1249:
1248:
1231:
1227:
1225:
1222:
1221:
1202:
1199:
1198:
1170:
1168:
1165:
1164:
1161:
1128:
1125:
1124:
1093:
1090:
1089:
1069:
1066:
1065:
1037:
1034:
1033:
996:
993:
992:
962:
959:
958:
937:
920:
917:
916:
898:
895:
894:
866:
863:
862:
838:
833:
816:
813:
812:
782:
779:
778:
747:
744:
743:
737:
724:Function Theory
720:Bernard Bolzano
716:
692:Rolle's theorem
657:
654:
653:
637:
634:
633:
617:
614:
613:
597:
594:
593:
526:
523:
522:
506:
503:
502:
486:
483:
482:
450:
447:
446:
430:
427:
426:
337:
334:
333:
305:
302:
301:
285:
282:
281:
265:
262:
261:
237:
234:
233:
205:
202:
201:
173:
170:
169:
126:
123:
122:
97:
94:
93:
82:
71:
65:
62:
52:Please help to
51:
35:
31:
24:
21:
12:
11:
5:
11335:
11325:
11324:
11319:
11302:
11301:
11299:
11298:
11293:
11288:
11283:
11278:
11276:Gabriel's horn
11273:
11268:
11267:
11266:
11261:
11256:
11251:
11246:
11238:
11237:
11236:
11227:
11225:
11221:
11220:
11217:
11216:
11214:
11213:
11208:
11206:List of limits
11202:
11199:
11198:
11196:
11195:
11194:
11193:
11188:
11183:
11173:
11172:
11171:
11161:
11156:
11151:
11146:
11140:
11138:
11129:
11125:
11124:
11122:
11121:
11114:
11107:
11105:Leonhard Euler
11102:
11097:
11092:
11087:
11082:
11077:
11072:
11067:
11062:
11057:
11051:
11049:
11043:
11042:
11040:
11039:
11034:
11029:
11024:
11019:
11013:
11011:
11005:
11004:
11002:
11001:
11000:
10999:
10994:
10989:
10984:
10979:
10974:
10969:
10964:
10959:
10954:
10946:
10945:
10944:
10939:
10938:
10937:
10932:
10922:
10917:
10912:
10907:
10902:
10897:
10889:
10883:
10881:
10877:
10876:
10874:
10873:
10872:
10871:
10866:
10861:
10856:
10848:
10843:
10838:
10833:
10828:
10823:
10818:
10813:
10808:
10806:Hessian matrix
10803:
10798:
10792:
10790:
10784:
10783:
10781:
10780:
10779:
10778:
10773:
10768:
10763:
10761:Line integrals
10755:
10754:
10753:
10748:
10743:
10738:
10733:
10724:
10722:
10716:
10715:
10713:
10712:
10707:
10702:
10701:
10700:
10695:
10687:
10682:
10681:
10680:
10670:
10669:
10668:
10663:
10658:
10648:
10643:
10642:
10641:
10631:
10626:
10621:
10616:
10611:
10609:Antiderivative
10605:
10603:
10597:
10596:
10594:
10593:
10592:
10591:
10586:
10581:
10571:
10570:
10569:
10564:
10556:
10555:
10554:
10549:
10544:
10539:
10529:
10528:
10527:
10522:
10517:
10512:
10504:
10503:
10502:
10497:
10496:
10495:
10485:
10480:
10475:
10470:
10465:
10455:
10454:
10453:
10448:
10438:
10433:
10428:
10423:
10418:
10413:
10407:
10405:
10399:
10398:
10396:
10395:
10390:
10385:
10380:
10379:
10378:
10368:
10362:
10360:
10354:
10353:
10351:
10350:
10345:
10340:
10335:
10330:
10325:
10320:
10315:
10310:
10305:
10300:
10295:
10290:
10284:
10282:
10276:
10275:
10268:
10267:
10260:
10253:
10245:
10239:
10238:
10229:
10210:
10200:
10189:
10188:External links
10186:
10185:
10184:
10178:
10158:Protter, M. H.
10154:
10148:
10133:
10130:
10127:
10126:
10119:
10096:
10089:
10066:
10055:(3): 303–311.
10038:
10037:
10035:
10032:
10007:If a function
9999:
9987:
9984:
9981:
9978:
9975:
9972:
9969:
9966:
9963:
9959:
9956:
9953:
9950:
9947:
9944:
9941:
9938:
9935:
9930:
9927:
9924:
9920:
9919:lim inf
9904:
9901:
9898:
9895:
9892:
9889:
9886:
9883:
9880:
9877:
9874:
9871:
9868:
9865:
9819:
9817:
9789:
9787:
9778:limit superior
9757:
9754:
9730:
9727:
9724:
9721:
9718:
9715:
9712:
9698:
9681:If a function
9673:
9661:
9658:
9655:
9652:
9649:
9646:
9643:
9640:
9637:
9633:
9630:
9627:
9624:
9621:
9618:
9615:
9612:
9609:
9604:
9601:
9598:
9594:
9593:lim sup
9578:
9575:
9572:
9569:
9566:
9563:
9560:
9557:
9554:
9551:
9548:
9545:
9542:
9539:
9512:
9509:
9507:
9506:
9491:
9471:
9468:
9465:
9462:
9459:
9439:
9417:
9413:
9409:
9406:
9403:
9400:
9397:
9394:
9391:
9369:
9365:
9344:
9340:
9336:
9314:
9310:
9287:
9283:
9279:
9276:
9273:
9270:
9267:
9264:
9261:
9241:
9238:
9235:
9232:
9229:
9209:
9189:
9186:
9183:
9180:
9177:
9174:
9171:
9168:
9165:
9160:
9156:
9135:
9132:
9129:
9126:
9123:
9120:
9117:
9114:
9094:
9074:
9071:
9068:
9048:
9028:
9008:
8988:
8985:
8982:
8979:
8976:
8973:
8970:
8950:
8946:
8942:
8939:
8936:
8916:
8896:
8893:
8890:
8887:
8884:
8881:
8878:
8858:
8838:
8834:
8830:
8827:
8823:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8798:
8795:
8791:
8770:
8767:
8764:
8744:
8724:
8703:
8700:
8697:
8686:
8674:
8654:
8651:
8648:
8645:
8642:
8622:
8619:
8616:
8613:
8610:
8607:
8604:
8601:
8598:
8595:
8575:
8572:
8569:
8566:
8563:
8560:
8557:
8537:
8515:
8511:
8507:
8504:
8501:
8498:
8495:
8492:
8489:
8467:
8463:
8442:
8438:
8434:
8412:
8408:
8385:
8381:
8377:
8374:
8371:
8368:
8365:
8362:
8359:
8339:
8336:
8333:
8330:
8327:
8307:
8287:
8284:
8281:
8278:
8275:
8272:
8269:
8266:
8263:
8258:
8254:
8233:
8230:
8227:
8224:
8221:
8218:
8215:
8212:
8192:
8172:
8169:
8166:
8146:
8126:
8106:
8086:
8083:
8080:
8077:
8074:
8071:
8068:
8065:
8062:
8042:
8038:
8034:
8031:
8028:
8008:
7988:
7985:
7982:
7979:
7976:
7973:
7970:
7967:
7964:
7944:
7924:
7920:
7916:
7913:
7909:
7905:
7902:
7899:
7896:
7893:
7890:
7887:
7884:
7881:
7877:
7856:
7853:
7850:
7830:
7810:
7789:
7786:
7783:
7759:
7756:
7753:
7750:
7747:
7744:
7741:
7738:
7718:
7715:
7712:
7709:
7706:
7703:
7680:
7677:
7674:
7671:
7668:
7665:
7645:
7625:
7605:
7602:
7599:
7579:
7559:
7556:
7553:
7550:
7547:
7527:
7507:
7484:
7464:
7461:
7458:
7455:
7452:
7449:
7446:
7426:
7422:
7418:
7415:
7412:
7392:
7372:
7369:
7366:
7363:
7360:
7357:
7354:
7334:
7314:
7310:
7306:
7303:
7299:
7295:
7292:
7289:
7286:
7283:
7280:
7277:
7274:
7271:
7267:
7246:
7243:
7240:
7220:
7200:
7177:
7157:
7137:
7134:
7131:
7128:
7125:
7122:
7102:
7082:
7062:
7042:
7022:
7019:
7016:
6996:
6993:
6990:
6967:
6964:
6961:
6958:
6955:
6952:
6949:
6946:
6926:
6923:
6920:
6917:
6914:
6894:
6874:
6854:
6851:
6848:
6845:
6842:
6839:
6836:
6833:
6813:
6810:
6807:
6804:
6801:
6798:
6778:
6775:
6772:
6769:
6766:
6763:
6740:
6720:
6700:
6680:
6660:
6657:
6654:
6651:
6631:
6628:
6625:
6622:
6619:
6616:
6596:
6576:
6573:
6570:
6567:
6564:
6561:
6541:
6538:
6535:
6515:
6512:
6509:
6506:
6503:
6483:
6463:
6460:
6457:
6454:
6451:
6448:
6428:
6408:
6405:
6402:
6399:
6396:
6376:
6373:
6370:
6367:
6364:
6344:
6341:
6338:
6335:
6321:
6309:
6306:
6303:
6300:
6297:
6277:
6274:
6271:
6268:
6265:
6245:
6242:
6239:
6236:
6221:
6218:
6182:
6181:
6169:
6166:
6163:
6160:
6157:
6154:
6151:
6144:
6140:
6135:
6131:
6127:
6124:
6120:
6117:
6114:
6111:
6108:
6101:
6097:
6092:
6088:
6083:
6079:
6075:
6071:
6068:
6040:
6037:
6032:
6028:
6024:
6019:
6015:
6011:
6007:
6004:
6000:
5997:
5994:
5987:
5983:
5978:
5974:
5969:
5965:
5961:
5957:
5954:
5933:
5928:
5924:
5920:
5915:
5911:
5907:
5904:
5897:
5893:
5888:
5884:
5879:
5875:
5853:
5835:
5830:
5827:
5824:
5820:
5816:
5811:
5807:
5803:
5800:
5797:
5765:
5758:
5754:
5749:
5745:
5741:
5738:
5734:
5731:
5703:
5698:
5694:
5690:
5685:
5681:
5677:
5674:
5667:
5663:
5658:
5654:
5649:
5645:
5628:
5621:
5609:
5566:
5534:
5532:
5529:
5519:) =
5511:in such that
5434:in such that
5389:) =
5365: on
5309:
5291:). Therefore,
5271:
5259:
5257:
5227:
5223:
5218:
5196:is in . Since
5173:
5169:
5164:
5139:
5120:
5093:
5086:
5067:
5028:in such that
5003:
5001:
4998:
4981:
4978:
4975:
4972:
4969:
4960:is bounded on
4949:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4891:
4888:
4885:
4882:
4879:
4870:is bounded on
4859:
4839:
4835:
4831:
4828:
4825:
4805:
4785:
4765:
4745:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4687:
4667:
4664:
4660:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4628:
4607:
4604:
4601:
4581:
4561:
4541:
4538:
4535:
4512:
4492:
4489:
4486:
4483:
4480:
4477:
4474:
4465:is bounded on
4454:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4390:
4387:
4384:
4381:
4378:
4369:is bounded on
4358:
4338:
4334:
4330:
4327:
4324:
4304:
4284:
4264:
4244:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4180:
4160:
4157:
4153:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4121:
4100:
4097:
4094:
4074:
4054:
4034:
4031:
4028:
4005:
4002:
3999:
3979:
3959:
3939:
3936:
3933:
3930:
3927:
3907:
3887:
3867:
3844:
3824:
3801:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3743:
3740:
3737:
3734:
3731:
3728:
3708:
3705:
3702:
3699:
3696:
3693:
3684:is bounded by
3673:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3615:
3595:
3592:
3588:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3556:
3535:
3532:
3529:
3509:
3489:
3466:
3446:
3426:
3406:
3386:
3366:
3363:
3360:
3340:
3337:
3334:
3331:
3311:
3308:
3305:
3302:
3299:
3290:is bounded on
3279:
3276:
3273:
3270:
3250:
3230:
3227:
3224:
3221:
3218:
3209:is bounded on
3198:
3195:
3192:
3189:
3169:
3166:
3163:
3160:
3157:
3137:
3117:
3103:
3086:
3083:
3080:
3077:
3074:
3054:
3033:
3030:
3009:
3002:
2997:
2991:
2987:
2984:
2964:
2944:
2941:
2936:
2932:
2928:
2925:
2918:
2913:
2907:
2903:
2900:
2880:
2856:
2836:
2833:
2830:
2827:
2807:
2800:
2795:
2789:
2785:
2782:
2762:
2742:
2722:
2702:
2699:
2696:
2693:
2690:
2670:
2650:
2644:
2640:
2635:
2612:
2608:
2605:
2601:
2593:
2589:
2584:
2580:
2556:
2553:
2550:
2547:
2544:
2521:
2517:
2514:
2510:
2504:
2500:
2496:
2473:
2470:
2467:
2462:
2458:
2454:
2451:
2431:
2428:
2425:
2422:
2419:
2416:
2411:
2407:
2386:
2366:
2363:
2360:
2357:
2354:
2334:
2320:
2307:
2304:
2301:
2298:
2295:
2292:
2272:
2269:
2266:
2263:
2260:
2257:
2237:
2234:
2231:
2228:
2211:
2209:
2206:
2205:
2204:
2201:
2190:
2178:
2159:
2136:
2116:
2113:
2093:
2077:
2074:
2053:
2050:
2047:
2044:
2039:
2036:
2033:
2029:
2025:
2022:
2019:
2016:
2013:
1993:
1990:
1987:
1984:
1979:
1976:
1973:
1969:
1965:
1962:
1959:
1956:
1953:
1933:
1930:
1927:
1924:
1921:
1901:
1880:
1876:
1873:
1870:
1867:
1847:
1830:
1817:
1789:
1769:
1749:
1746:
1743:
1740:
1719:
1715:
1712:
1688:
1685:
1682:
1679:
1676:
1673:
1653:
1650:
1647:
1627:
1624:
1621:
1618:
1615:
1595:
1589:
1586:
1569:
1556:
1553:
1550:
1547:
1544:
1539:
1536:
1532:
1511:
1508:
1505:
1485:
1482:
1479:
1476:
1473:
1453:
1447:
1444:
1413:
1393:
1390:
1383:
1379:
1374:
1368:
1363:
1360:
1357:
1353:
1328:
1324:
1319:
1315:
1312:
1309:
1302:
1298:
1293:
1272:
1269:
1264:
1260:
1256:
1234:
1230:
1206:
1173:
1160:
1157:
1144:
1141:
1138:
1135:
1132:
1112:
1109:
1106:
1103:
1100:
1097:
1086:
1085:
1073:
1053:
1050:
1047:
1044:
1041:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
990:
978:
975:
972:
969:
966:
944:
941:
936:
933:
930:
927:
924:
914:
902:
882:
879:
876:
873:
870:
847:
844:
841:
837:
832:
829:
826:
823:
820:
810:
798:
795:
792:
789:
786:
766:
763:
760:
757:
754:
751:
736:
733:
715:
712:
676:
673:
670:
667:
664:
661:
641:
621:
601:
579:
576:
573:
570:
567:
564:
561:
558:
555:
551:
548:
545:
542:
539:
536:
533:
530:
510:
490:
466:
463:
460:
457:
454:
434:
408:
405:
402:
399:
396:
393:
390:
387:
384:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
321:
318:
315:
312:
309:
289:
269:
252:must attain a
241:
221:
218:
215:
212:
209:
177:
142:
139:
136:
133:
130:
110:
107:
104:
101:
84:
83:
38:
36:
29:
22:
9:
6:
4:
3:
2:
11334:
11323:
11320:
11318:
11315:
11314:
11312:
11297:
11294:
11292:
11289:
11287:
11284:
11282:
11279:
11277:
11274:
11272:
11269:
11265:
11262:
11260:
11257:
11255:
11252:
11250:
11247:
11245:
11242:
11241:
11239:
11235:
11232:
11231:
11229:
11228:
11226:
11222:
11212:
11209:
11207:
11204:
11203:
11200:
11192:
11189:
11187:
11184:
11182:
11179:
11178:
11177:
11174:
11170:
11167:
11166:
11165:
11162:
11160:
11157:
11155:
11152:
11150:
11147:
11145:
11142:
11141:
11139:
11137:
11133:
11130:
11126:
11120:
11119:
11115:
11113:
11112:
11108:
11106:
11103:
11101:
11098:
11096:
11093:
11091:
11088:
11086:
11083:
11081:
11080:Infinitesimal
11078:
11076:
11073:
11071:
11068:
11066:
11063:
11061:
11058:
11056:
11053:
11052:
11050:
11048:
11044:
11038:
11035:
11033:
11030:
11028:
11025:
11023:
11020:
11018:
11015:
11014:
11012:
11006:
10998:
10995:
10993:
10990:
10988:
10985:
10983:
10980:
10978:
10975:
10973:
10970:
10968:
10965:
10963:
10960:
10958:
10955:
10953:
10950:
10949:
10947:
10943:
10940:
10936:
10933:
10931:
10928:
10927:
10926:
10923:
10921:
10918:
10916:
10913:
10911:
10908:
10906:
10903:
10901:
10898:
10896:
10893:
10892:
10890:
10888:
10885:
10884:
10882:
10878:
10870:
10867:
10865:
10862:
10860:
10857:
10855:
10852:
10851:
10849:
10847:
10844:
10842:
10839:
10837:
10834:
10832:
10829:
10827:
10824:
10822:
10821:Line integral
10819:
10817:
10814:
10812:
10809:
10807:
10804:
10802:
10799:
10797:
10794:
10793:
10791:
10789:
10785:
10777:
10774:
10772:
10769:
10767:
10764:
10762:
10759:
10758:
10756:
10752:
10749:
10747:
10744:
10742:
10739:
10737:
10734:
10732:
10729:
10728:
10726:
10725:
10723:
10721:
10717:
10711:
10708:
10706:
10703:
10699:
10696:
10694:
10693:Washer method
10691:
10690:
10688:
10686:
10683:
10679:
10676:
10675:
10674:
10671:
10667:
10664:
10662:
10659:
10657:
10656:trigonometric
10654:
10653:
10652:
10649:
10647:
10644:
10640:
10637:
10636:
10635:
10632:
10630:
10627:
10625:
10622:
10620:
10617:
10615:
10612:
10610:
10607:
10606:
10604:
10602:
10598:
10590:
10587:
10585:
10582:
10580:
10577:
10576:
10575:
10572:
10568:
10565:
10563:
10560:
10559:
10557:
10553:
10550:
10548:
10545:
10543:
10540:
10538:
10535:
10534:
10533:
10530:
10526:
10525:Related rates
10523:
10521:
10518:
10516:
10513:
10511:
10508:
10507:
10505:
10501:
10498:
10494:
10491:
10490:
10489:
10486:
10484:
10481:
10479:
10476:
10474:
10471:
10469:
10466:
10464:
10461:
10460:
10459:
10456:
10452:
10449:
10447:
10444:
10443:
10442:
10439:
10437:
10434:
10432:
10429:
10427:
10424:
10422:
10419:
10417:
10414:
10412:
10409:
10408:
10406:
10404:
10400:
10394:
10391:
10389:
10386:
10384:
10381:
10377:
10374:
10373:
10372:
10369:
10367:
10364:
10363:
10361:
10359:
10355:
10349:
10346:
10344:
10341:
10339:
10336:
10334:
10331:
10329:
10326:
10324:
10321:
10319:
10316:
10314:
10311:
10309:
10306:
10304:
10301:
10299:
10296:
10294:
10291:
10289:
10286:
10285:
10283:
10281:
10277:
10273:
10266:
10261:
10259:
10254:
10252:
10247:
10246:
10243:
10237:
10233:
10230:
10225:
10224:
10219:
10216:
10211:
10208:
10204:
10201:
10199:
10195:
10192:
10191:
10181:
10179:0-387-90215-5
10175:
10171:
10167:
10163:
10162:Morrey, C. B.
10159:
10155:
10151:
10149:0-201-82823-5
10145:
10141:
10136:
10135:
10122:
10120:0-87150-911-3
10116:
10109:
10108:
10100:
10092:
10090:0-07-054235-X
10086:
10082:
10081:
10073:
10071:
10062:
10058:
10054:
10050:
10043:
10039:
10031:
10026:
10024:
10020:
10011:
9998:
9985:
9979:
9976:
9973:
9967:
9964:
9954:
9948:
9945:
9939:
9933:
9928:
9922:
9896:
9890:
9878:
9875:
9872:
9866:
9863:
9855:
9848:
9847:
9843:
9839:
9835:
9831:
9827:
9823:
9813:
9809:
9805:
9801:
9797:
9793:
9783:
9779:
9775:
9771:
9752:
9744:
9725:
9722:
9716:
9710:
9696:
9694:
9685:
9672:
9659:
9653:
9650:
9647:
9641:
9638:
9628:
9622:
9619:
9613:
9607:
9602:
9596:
9570:
9564:
9552:
9549:
9546:
9540:
9537:
9528:
9526:
9522:
9518:
9505:
9489:
9466:
9463:
9460:
9437:
9415:
9411:
9407:
9404:
9401:
9395:
9389:
9367:
9363:
9342:
9338:
9334:
9312:
9308:
9285:
9281:
9277:
9274:
9271:
9265:
9259:
9236:
9233:
9230:
9207:
9200:then for all
9184:
9181:
9178:
9172:
9169:
9166:
9163:
9158:
9154:
9133:
9130:
9124:
9121:
9118:
9112:
9092:
9072:
9069:
9066:
9046:
9026:
9006:
8983:
8980:
8977:
8974:
8971:
8948:
8944:
8940:
8937:
8934:
8927:is less than
8914:
8891:
8888:
8885:
8882:
8879:
8856:
8836:
8832:
8828:
8825:
8814:
8808:
8805:
8799:
8793:
8768:
8765:
8762:
8742:
8722:
8714:
8701:
8698:
8695:
8687:
8672:
8652:
8649:
8646:
8643:
8640:
8620:
8617:
8611:
8608:
8605:
8602:
8599:
8593:
8570:
8567:
8564:
8561:
8558:
8535:
8513:
8509:
8505:
8502:
8499:
8493:
8487:
8465:
8461:
8440:
8436:
8432:
8410:
8406:
8383:
8379:
8375:
8372:
8369:
8363:
8357:
8334:
8331:
8328:
8305:
8298:then for all
8282:
8279:
8276:
8270:
8267:
8264:
8261:
8256:
8252:
8231:
8228:
8222:
8219:
8216:
8210:
8190:
8170:
8167:
8164:
8144:
8124:
8104:
8081:
8078:
8075:
8072:
8069:
8066:
8063:
8040:
8036:
8032:
8029:
8026:
8019:is less than
8006:
7983:
7980:
7977:
7974:
7971:
7968:
7965:
7942:
7922:
7918:
7914:
7911:
7900:
7894:
7891:
7885:
7879:
7854:
7851:
7848:
7828:
7808:
7800:
7787:
7784:
7781:
7773:
7772:
7771:
7754:
7748:
7745:
7742:
7739:
7736:
7716:
7713:
7707:
7701:
7692:
7678:
7675:
7669:
7663:
7643:
7623:
7603:
7600:
7597:
7577:
7554:
7551:
7548:
7525:
7505:
7496:
7482:
7459:
7456:
7453:
7450:
7447:
7424:
7420:
7416:
7413:
7410:
7403:is less than
7390:
7367:
7364:
7361:
7358:
7355:
7332:
7312:
7308:
7304:
7301:
7290:
7284:
7281:
7275:
7269:
7244:
7241:
7238:
7218:
7198:
7189:
7175:
7155:
7132:
7129:
7126:
7120:
7100:
7080:
7060:
7040:
7020:
7017:
7014:
6994:
6991:
6988:
6979:
6965:
6962:
6956:
6953:
6950:
6944:
6921:
6918:
6915:
6892:
6872:
6849:
6843:
6840:
6837:
6834:
6831:
6811:
6808:
6802:
6796:
6776:
6773:
6767:
6761:
6752:
6738:
6718:
6698:
6678:
6655:
6649:
6626:
6623:
6620:
6614:
6594:
6571:
6568:
6565:
6559:
6539:
6536:
6533:
6510:
6507:
6504:
6481:
6458:
6455:
6452:
6446:
6426:
6403:
6400:
6397:
6371:
6368:
6365:
6339:
6333:
6320:
6304:
6301:
6298:
6272:
6269:
6266:
6240:
6234:
6226:
6216:
6215:
6211:
6207:
6203:
6199:
6195:
6191:
6187:
6164:
6158:
6155:
6142:
6138:
6133:
6115:
6112:
6099:
6095:
6090:
6081:
6077:
6058:
6057:
6056:
6054:
6030:
6026:
6017:
6013:
5998:
5985:
5981:
5976:
5967:
5963:
5926:
5922:
5913:
5909:
5905:
5895:
5891:
5886:
5877:
5873:
5864:
5860:
5856:
5849:
5828:
5825:
5822:
5818:
5814:
5809:
5805:
5798:
5795:
5787:
5783:
5779:
5756:
5752:
5747:
5732:
5729:
5721:
5717:
5696:
5692:
5683:
5679:
5675:
5665:
5661:
5656:
5647:
5643:
5634:
5627:
5620:
5616:
5612:
5605:
5601:
5597:
5593:
5589:
5585:
5581:
5577:
5573:
5569:
5562:
5558:
5557:infinitesimal
5554:
5550:
5546:
5542:
5527:
5526:
5522:
5518:
5514:
5510:
5504:
5500:
5496:
5488:
5484:
5480:
5473:
5469:
5465:
5461:
5455:
5450:
5446:
5442:
5438:
5433:
5429:
5424:
5420:
5416:
5412:
5405:
5401:
5397:
5392:
5388:
5384:
5380:
5374:
5370:
5362:
5358:
5354:
5349:
5345:
5339:
5335:
5331:
5327:
5323:
5319:
5307:
5306:
5302:
5298:
5294:
5290:
5286:
5282:
5278:
5274:
5267:
5263:
5253:
5249:
5245:
5225:
5221:
5216:
5207:
5203:
5199:
5195:
5191:
5171:
5167:
5162:
5153:
5148:
5146:
5142:
5135:
5131:
5127:
5123:
5116:
5112:
5108:
5104:
5100:
5096:
5089:
5082:
5078:
5074:
5070:
5063:
5059:
5055:
5052:upper bound,
5051:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5019:
5015:
5011:
4996:
4995:
4976:
4973:
4970:
4947:
4924:
4921:
4918:
4915:
4912:
4886:
4883:
4880:
4857:
4837:
4833:
4829:
4826:
4823:
4803:
4783:
4763:
4743:
4720:
4717:
4714:
4711:
4708:
4685:
4665:
4662:
4651:
4645:
4642:
4636:
4630:
4605:
4602:
4599:
4579:
4559:
4539:
4536:
4533:
4524:
4510:
4487:
4484:
4481:
4478:
4475:
4452:
4429:
4426:
4423:
4420:
4417:
4414:
4411:
4385:
4382:
4379:
4356:
4336:
4332:
4328:
4325:
4322:
4302:
4282:
4262:
4242:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4178:
4158:
4155:
4144:
4138:
4135:
4129:
4123:
4098:
4095:
4092:
4072:
4052:
4032:
4029:
4026:
4017:
4003:
4000:
3997:
3977:
3957:
3934:
3931:
3928:
3905:
3885:
3865:
3856:
3842:
3822:
3813:
3799:
3776:
3773:
3770:
3767:
3764:
3741:
3738:
3732:
3726:
3706:
3703:
3697:
3691:
3671:
3648:
3645:
3642:
3639:
3636:
3613:
3593:
3590:
3579:
3573:
3570:
3564:
3558:
3533:
3530:
3527:
3507:
3487:
3478:
3464:
3444:
3424:
3404:
3384:
3364:
3361:
3358:
3335:
3329:
3322:by the value
3306:
3303:
3300:
3274:
3268:
3248:
3225:
3222:
3219:
3193:
3187:
3164:
3161:
3158:
3135:
3115:
3101:
3100:
3081:
3078:
3075:
3052:
3044:
3028:
3000:
2995:
2989:
2982:
2962:
2942:
2939:
2934:
2930:
2926:
2916:
2911:
2905:
2898:
2878:
2870:
2854:
2831:
2825:
2798:
2793:
2787:
2780:
2760:
2740:
2720:
2697:
2694:
2691:
2668:
2642:
2638:
2606:
2603:
2591:
2587:
2582:
2570:
2551:
2548:
2545:
2515:
2512:
2502:
2498:
2487:
2471:
2468:
2460:
2456:
2449:
2426:
2423:
2420:
2414:
2409:
2405:
2384:
2361:
2358:
2355:
2332:
2318:
2305:
2299:
2296:
2293:
2270:
2264:
2261:
2258:
2232:
2226:
2202:
2199:
2195:
2191:
2176:
2168:
2164:
2160:
2157:
2156:
2155:
2152:
2150:
2134:
2114:
2111:
2091:
2083:
2073:
2071:
2065:
2048:
2042:
2037:
2034:
2031:
2023:
2017:
2011:
1988:
1982:
1977:
1974:
1971:
1963:
1957:
1951:
1931:
1928:
1925:
1922:
1919:
1899:
1871:
1868:
1865:
1845:
1829:
1815:
1807:
1803:
1787:
1767:
1744:
1738:
1713:
1710:
1700:
1686:
1683:
1677:
1671:
1651:
1648:
1645:
1625:
1619:
1616:
1613:
1593:
1587:
1584:
1568:
1554:
1551:
1545:
1537:
1534:
1530:
1509:
1506:
1503:
1483:
1477:
1474:
1471:
1464:, a function
1451:
1445:
1442:
1433:
1431:
1427:
1411:
1391:
1388:
1381:
1377:
1372:
1366:
1361:
1358:
1355:
1351:
1326:
1322:
1317:
1313:
1310:
1307:
1300:
1296:
1291:
1270:
1267:
1262:
1258:
1254:
1232:
1228:
1220:
1204:
1196:
1192:
1188:
1187:metric spaces
1156:
1139:
1136:
1133:
1110:
1107:
1101:
1095:
1071:
1048:
1045:
1042:
1032:defined over
1019:
1016:
1013:
1010:
1004:
998:
991:
973:
970:
967:
957:defined over
942:
939:
934:
928:
922:
915:
900:
874:
871:
861:defined over
845:
842:
839:
835:
830:
824:
818:
811:
790:
787:
777:defined over
764:
761:
755:
749:
742:
741:
740:
732:
730:
725:
721:
711:
709:
705:
701:
700:compact space
697:
693:
688:
674:
668:
665:
662:
639:
619:
599:
590:
577:
571:
568:
565:
559:
556:
549:
546:
540:
534:
531:
528:
508:
488:
480:
461:
458:
455:
432:
424:
419:
406:
400:
397:
394:
388:
385:
375:
369:
366:
360:
354:
351:
345:
339:
316:
313:
310:
287:
267:
259:
255:
239:
216:
213:
210:
199:
195:
191:
175:
168:
164:
160:
137:
134:
131:
105:
99:
90:
80:
77:
69:
59:
55:
49:
48:
42:
37:
28:
27:
19:
11191:Secant cubed
11116:
11109:
11090:Isaac Newton
11060:Brook Taylor
10727:Derivatives
10698:Shell method
10546:
10426:Differential
10232:Mizar system
10221:
10198:cut-the-knot
10169:
10139:
10106:
10099:
10079:
10052:
10048:
10042:
10028:
10018:
10009:
10000:
9853:
9850:
9841:
9837:
9833:
9829:
9825:
9815:
9811:
9807:
9803:
9799:
9795:
9785:
9781:
9773:
9769:
9742:
9702:
9692:
9683:
9674:
9529:
9516:
9514:
8688:
7774:
7693:
7497:
7190:
6980:
6753:
6325:
6224:
6223:
6209:
6205:
6201:
6197:
6193:
6189:
6185:
6183:
6052:
5944:, we obtain
5862:
5861:. Applying
5858:
5851:
5847:
5785:
5777:
5719:
5715:
5632:
5625:
5618:
5607:
5603:
5599:
5595:
5591:
5587:
5583:
5579:
5575:
5571:
5564:
5560:
5552:
5549:hyperinteger
5544:
5538:
5520:
5516:
5512:
5508:
5502:
5498:
5494:
5486:
5482:
5478:
5471:
5467:
5463:
5459:
5453:
5448:
5444:
5440:
5436:
5431:
5427:
5425:
5418:
5414:
5410:
5403:
5399:
5395:
5390:
5386:
5382:
5381:on so that
5378:
5372:
5368:
5360:
5356:
5352:
5337:
5333:
5329:
5325:
5321:
5317:
5313:
5300:
5296:
5292:
5288:
5284:
5280:
5276:
5269:
5265:
5255:
5251:
5247:
5243:
5205:
5201:
5197:
5193:
5189:
5149:
5144:
5137:
5133:
5129:
5125:
5118:
5114:
5110:
5106:
5102:
5098:
5091:
5084:
5080:
5076:
5072:
5071:in so that
5065:
5061:
5057:
5053:
5049:
5045:
5041:
5037:
5033:
5029:
5025:
5021:
5017:
5009:
5007:
4525:
4018:
3857:
3814:
3479:
3107:
3020:diverges to
2324:
2212:
2153:
2149:real numbers
2079:
2067:
1831:
1800:attains its
1702:
1570:
1434:
1189:and general
1162:
1087:
738:
723:
717:
708:real numbers
689:
591:
422:
420:
162:
156:
72:
63:
44:
11259:of surfaces
11010:and numbers
10972:Dirichlet's
10942:Telescoping
10895:Alternating
10483:L'Hôpital's
10280:Precalculus
10014:(–∞, ∞]
10012: : →
9688:[–∞, ∞)
9686: : →
9530:A function
6607:, and that
5336:) for some
2733:. Because
2194:subsequence
2082:upper bound
1247:such that
1195:compact set
521:such that:
332:such that:
58:introducing
11311:Categories
11055:Adequality
10741:Divergence
10614:Arc length
10411:Derivative
10034:References
9382:, we have
9300:. Taking
8781:such that
8480:, we have
8398:. Taking
7867:such that
7257:such that
6937:such that
6885:of points
6204:, proving
5606:+1 points
5470:)) > 1/
5346:exists by
5105:, we have
4618:such that
4111:such that
3546:such that
3180:such that
3128:of points
2955:for every
2535:. Because
2442:such that
1944:such that
190:continuous
41:references
11254:of curves
11249:Curvature
11136:Integrals
10930:Maclaurin
10910:Geometric
10801:Geometric
10751:Laplacian
10463:linearity
10303:Factorial
10223:MathWorld
9968:∈
9962:∀
9946:≥
9926:→
9900:∞
9894:∞
9891:−
9885:→
9756:∞
9753:−
9729:∞
9726:−
9642:∈
9636:∀
9620:≤
9600:→
9574:∞
9568:∞
9565:−
9559:→
9408:−
9402:≤
9278:−
9272:≤
9170:−
9073:δ
9070:−
8978:δ
8975:−
8938:−
8886:δ
8883:−
8806:−
8763:δ
8650:∈
8647:δ
8612:δ
8586:. Hence
8571:δ
8506:−
8500:≤
8376:−
8370:≤
8268:−
8171:δ
8168:−
8082:δ
8070:δ
8067:−
8030:−
7984:δ
7972:δ
7969:−
7892:−
7849:δ
7746:−
7460:δ
7414:−
7368:δ
7282:−
7239:δ
6992:∈
6981:Clearly
6841:−
6537:≤
6225:Statement
6192:) ≥
6082:∗
6018:∗
5999:≥
5968:∗
5914:∗
5906:≥
5878:∗
5857:) =
5799:∈
5684:∗
5676:≥
5648:∗
5559:length 1/
5340:∈ }
5097:}. Since
4919:δ
4916:−
4830:δ
4827:−
4715:δ
4712:−
4643:−
4600:δ
4488:δ
4430:δ
4418:δ
4415:−
4329:δ
4326:−
4220:δ
4208:δ
4205:−
4136:−
4093:δ
3777:δ
3704:−
3649:δ
3571:−
3528:δ
3032:∞
2940:≥
2607:∈
2516:∈
2415:∈
2112:−
2035:∈
1975:∈
1929:∈
1875:→
1684:⊂
1649:⊂
1623:→
1552:⊂
1535:−
1507:⊂
1481:→
1389:⊃
1378:α
1352:⋃
1323:α
1311:…
1297:α
1268:⊃
1263:α
1255:⋃
1233:α
1219:open sets
1088:Defining
1017:−
878:∞
794:∞
560:∈
554:∀
547:≤
532:≤
389:∈
383:∀
367:≤
352:≤
200:interval
66:June 2012
11244:Manifold
10977:Integral
10920:Infinite
10915:Harmonic
10900:Binomial
10746:Gradient
10689:Volumes
10500:Quotient
10441:Notation
10272:Calculus
10164:(1977).
9741:for all
9430:for all
9146:. Let
8849:for all
8633:so that
8528:for all
8244:. Let
7935:for all
7325:for all
7113:because
6824:and let
5631: ≤
5570: =
5497:−
5481:−
5462:−
5452:because
5439:−
5413:−
5324: :
5320:∈
5314:The set
5303:.
5250:). But {
5128:for all
4940:so that
4850:. Thus
4736:so that
4678:for all
4445:so that
4349:. Thus
4235:so that
4171:for all
4019:Suppose
3606:for all
2486:sequence
2167:supremum
1802:supremum
1197:. A set
167:function
159:calculus
11181:inverse
11169:inverse
11095:Fluxion
10905:Fourier
10771:Stokes'
10766:Green's
10488:Product
10348:Tangent
10234:proof:
10023:infimum
10002:Theorem
9844:.
9676:Theorem
7729:. Let
7383:. Thus
5780:is the
5574: /
5447:) <
5402:) <
5393:, then
5048:is the
5040:). Let
4552:. Now
4045:. Now
3664:. Thus
3097:.
2891:). But
1833:Theorem
1806:infimum
1572:Theorem
714:History
706:of the
479:bounded
258:minimum
254:maximum
232:, then
198:bounded
192:on the
54:improve
11264:Tensor
11186:Secant
10952:Abel's
10935:Taylor
10826:Matrix
10776:Gauss'
10358:Limits
10338:Secant
10328:Radian
10176:
10146:
10117:
10087:
7498:Next,
6526:where
6210:ƒ
6194:ƒ
6186:ƒ
6184:Hence
6053:ƒ
5776:where
5600:ƒ
5592:ƒ
5588:ƒ
5543:, let
5472:ε
5449:ε
5428:ε
5355:= sup(
3858:Next,
3351:. If
2198:domain
1591:
1449:
704:subset
256:and a
194:closed
161:, the
43:, but
11128:Lists
10987:Ratio
10925:Power
10661:Euler
10478:Chain
10468:Power
10343:Slope
10111:(PDF)
9700:Proof
6439:, or
6323:Proof
5536:Proof
5375:]
5367:[
5279:, so
5113:<
5079:<
5050:least
2681:. As
2322:Proof
2163:image
702:to a
10997:Term
10992:Root
10731:Curl
10174:ISBN
10144:ISBN
10115:ISBN
10085:ISBN
9840:) =
9355:and
9131:<
8826:<
8766:>
8618:<
8453:and
8229:<
7912:<
7852:>
7785:<
7714:<
7601:>
7302:<
7242:>
7191:Now
7073:and
7018:>
6963:<
6809:<
5635:and
5150:The
5124:) ≤
5109:– 1/
5075:– 1/
5056:– 1/
4663:<
4603:>
4156:<
4096:>
4030:<
4001:>
3719:and
3591:<
3531:>
3480:Now
3397:and
3362:>
2927:>
2847:(as
2469:>
2004:and
1804:and
612:and
501:and
280:and
196:and
10473:Sum
10196:at
10057:doi
9806:at
9772:at
9703:If
9450:in
9220:in
8869:in
8548:in
8318:in
7955:in
7345:in
6905:in
6754:If
6731:to
6691:as
6671:to
5578:as
5493:1/(
5477:1/(
5458:1/(
5409:1/(
5299:at
5020:of
4698:in
4191:in
3626:in
3148:in
2871:at
2867:is
2626:of
2219:If
2169:of
2072:).
2028:inf
1968:sup
1838:If
1577:If
1185:to
477:is
300:in
188:is
157:In
11313::
10220:.
10168:.
10160:;
10069:^
10053:32
10051:.
10025:.
9252:,
9105:,
8350:,
8203:,
7691:.
7495:.
7188:.
6978:.
6751:.
6212:.
5863:st
5848:st
5778:st
5523:.
5505:))
5489:))
5421:))
5371:,
5363:))
5328:=
5283:=
5147:.
5032:=
4796:,
4523:.
4295:,
4016:.
3855:.
3812:.
3477:.
2151:.
2064:.
1522:,
1155:.
731:.
10264:e
10257:t
10250:v
10226:.
10209:.
10182:.
10152:.
10123:.
10093:.
10063:.
10059::
10019:f
10010:f
9986:.
9983:]
9980:b
9977:,
9974:a
9971:[
9965:x
9958:)
9955:x
9952:(
9949:f
9943:)
9940:y
9937:(
9934:f
9929:x
9923:y
9903:)
9897:,
9888:[
9882:]
9879:b
9876:,
9873:a
9870:[
9867::
9864:f
9854:f
9846:∎
9842:M
9838:d
9836:(
9834:f
9830:d
9828:(
9826:f
9820:k
9818:n
9816:d
9814:(
9812:f
9808:d
9804:f
9800:x
9798:(
9796:f
9790:k
9788:n
9786:x
9784:(
9782:f
9774:x
9770:f
9743:x
9723:=
9720:)
9717:x
9714:(
9711:f
9693:f
9684:f
9660:.
9657:]
9654:b
9651:,
9648:a
9645:[
9639:x
9632:)
9629:x
9626:(
9623:f
9617:)
9614:y
9611:(
9608:f
9603:x
9597:y
9577:)
9571:,
9562:[
9556:]
9553:b
9550:,
9547:a
9544:[
9541::
9538:f
9517:f
9504:∎
9490:M
9470:]
9467:b
9464:,
9461:a
9458:[
9438:x
9416:2
9412:d
9405:M
9399:)
9396:x
9393:(
9390:f
9368:1
9364:d
9343:2
9339:/
9335:d
9313:2
9309:d
9286:1
9282:d
9275:M
9269:)
9266:x
9263:(
9260:f
9240:]
9237:e
9234:,
9231:a
9228:[
9208:x
9188:]
9185:e
9182:,
9179:a
9176:[
9173:M
9167:M
9164:=
9159:1
9155:d
9134:M
9128:]
9125:e
9122:,
9119:a
9116:[
9113:M
9093:L
9067:s
9047:L
9027:e
9007:s
8987:]
8984:s
8981:,
8972:s
8969:[
8949:2
8945:/
8941:d
8935:M
8915:f
8895:]
8892:s
8889:,
8880:s
8877:[
8857:x
8837:2
8833:/
8829:d
8822:|
8818:)
8815:s
8812:(
8809:f
8803:)
8800:x
8797:(
8794:f
8790:|
8769:0
8743:s
8723:f
8702:b
8699:=
8696:s
8673:s
8653:L
8644:+
8641:s
8621:M
8615:]
8609:+
8606:s
8603:,
8600:a
8597:[
8594:M
8574:]
8568:+
8565:s
8562:,
8559:a
8556:[
8536:x
8514:2
8510:d
8503:M
8497:)
8494:x
8491:(
8488:f
8466:1
8462:d
8441:2
8437:/
8433:d
8411:2
8407:d
8384:1
8380:d
8373:M
8367:)
8364:x
8361:(
8358:f
8338:]
8335:e
8332:,
8329:a
8326:[
8306:x
8286:]
8283:e
8280:,
8277:a
8274:[
8271:M
8265:M
8262:=
8257:1
8253:d
8232:M
8226:]
8223:e
8220:,
8217:a
8214:[
8211:M
8191:L
8165:s
8145:L
8125:e
8105:s
8085:]
8079:+
8076:s
8073:,
8064:s
8061:[
8041:2
8037:/
8033:d
8027:M
8007:f
7987:]
7981:+
7978:s
7975:,
7966:s
7963:[
7943:x
7923:2
7919:/
7915:d
7908:|
7904:)
7901:s
7898:(
7895:f
7889:)
7886:x
7883:(
7880:f
7876:|
7855:0
7829:s
7809:f
7788:b
7782:s
7758:)
7755:s
7752:(
7749:f
7743:M
7740:=
7737:d
7717:M
7711:)
7708:s
7705:(
7702:f
7679:M
7676:=
7673:)
7670:s
7667:(
7664:f
7644:f
7624:s
7604:a
7598:s
7578:s
7558:]
7555:b
7552:,
7549:a
7546:[
7526:b
7506:L
7483:L
7463:]
7457:+
7454:a
7451:,
7448:a
7445:[
7425:2
7421:/
7417:d
7411:M
7391:f
7371:]
7365:+
7362:a
7359:,
7356:a
7353:[
7333:x
7313:2
7309:/
7305:d
7298:|
7294:)
7291:a
7288:(
7285:f
7279:)
7276:x
7273:(
7270:f
7266:|
7245:0
7219:a
7199:f
7176:a
7156:L
7136:]
7133:x
7130:,
7127:a
7124:[
7121:M
7101:L
7081:e
7061:a
7041:L
7021:a
7015:e
6995:L
6989:a
6966:M
6960:]
6957:x
6954:,
6951:a
6948:[
6945:M
6925:]
6922:b
6919:,
6916:a
6913:[
6893:x
6873:L
6853:)
6850:a
6847:(
6844:f
6838:M
6835:=
6832:d
6812:M
6806:)
6803:a
6800:(
6797:f
6777:M
6774:=
6771:)
6768:a
6765:(
6762:f
6739:b
6719:a
6699:x
6679:M
6659:)
6656:a
6653:(
6650:f
6630:]
6627:x
6624:,
6621:a
6618:[
6615:M
6595:M
6575:]
6572:x
6569:,
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