803:
811:
717:
4189:
4169:
689:
31:
2032:
1861:
2440:
1295:
1180:
2246:
2920:
789:
in the space of all continuous functions. Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the
2659:
2027:{\displaystyle \lim _{\mathbf {h} \to \mathbf {0} }{\frac {\|\mathbf {f} (\mathbf {x_{0}} +\mathbf {h} )-\mathbf {f} (\mathbf {x_{0}} )-\mathbf {J} \mathbf {(h)} \|_{\mathbf {R} ^{n}}}{\|\mathbf {h} \|_{\mathbf {R} ^{m}}}}=0.}
1706:
366:
1185:
591:
916:
2261:
1064:
2774:
3125:
3016:
1423:
2142:
848:
2959:
2698:
2515:
1451:
Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless,
435:
463:
152:
654:
1798:
1051:
4063:
2782:
3060:
1324:
670:
if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section
1739:
1449:
942:
1496:
977:
493:
1590:
1556:
1526:
250:
2549:
1611:
271:
2724:
2541:
1762:
1007:
386:
218:
2664:
Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function
3311:
3905:
2477:, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing
1058:
1616:
276:
1332:
2435:{\displaystyle f(x,y)={\begin{cases}y^{3}/(x^{2}+y^{2})&{\text{if }}(x,y)\neq (0,0)\\0&{\text{if }}(x,y)=(0,0)\end{cases}}}
501:
3421:
853:
3304:
2255:, but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function
2729:
1290:{\displaystyle f'(0)=\lim _{\varepsilon \to 0}\left({\frac {\varepsilon ^{2}\sin(1/\varepsilon )-0}{\varepsilon }}\right)=0}
4094:
3147:
3065:
73:
at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a
4195:
3746:
3483:
2964:
1809:
4007:
3634:
3441:
3297:
3221:
17:
817:
3962:
1175:{\displaystyle f(x)\;=\;{\begin{cases}x^{2}\sin(1/x)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}
4228:
1452:
2928:
2667:
2484:
4223:
4149:
4089:
3687:
2457:
3682:
3371:
1053:
exists and is itself a continuous function. Although the derivative of a differentiable function never has a
802:
395:, the differentiability of it is something more complex than the existence of the partial derivatives of it.
4124:
3521:
3478:
3431:
3426:
4175:
3471:
3397:
2961:
can be differentiable as a multi-variable function, while not being complex-differentiable. For example,
2241:{\displaystyle f(x,y)={\begin{cases}x&{\text{if }}y\neq x^{2}\\0&{\text{if }}y=x^{2}\end{cases}}}
2085:
2078:
406:
3799:
3734:
3335:
3231:
1456:
440:
658:
A continuous function is not necessarily differentiable, but a differentiable function is necessarily
4233:
4200:
4058:
3697:
3528:
3351:
2291:
2172:
1090:
4099:
3356:
2915:{\displaystyle \lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {|f(a+h)-f(a)-f'(a)h|}{|h|}}=0.}
2077:
matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the
634:
1773:
724:
on the graph of a continuous function. At zero, the function is continuous but not differentiable.
4144:
4129:
3782:
3777:
3677:
3545:
3157:
1020:
4104:
3864:
3583:
3578:
3226:
2453:
2119:
762:. In particular, any differentiable function must be continuous at every point in its domain.
116:
51:
3024:
1300:
4134:
4119:
4084:
3772:
3672:
3540:
2122:) does not guarantee that a function is differentiable at a point. For example, the function
1711:
1428:
1326:
921:
4002:
2654:{\displaystyle f'(a)=\lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {f(a+h)-f(a)}{h}}.}
1471:
947:
472:
4154:
4109:
3555:
3500:
3346:
3341:
3131:
3019:
2468:
1568:
1563:
of the function both exist and are continuous. More generally, a function is said to be of
1534:
1504:
791:
228:
59:
8:
3729:
3707:
3456:
3451:
3409:
3361:
1463:
766:: a continuous function need not be differentiable. For example, a function with a bend,
750:
683:
659:
597:
63:
4114:
3692:
3257:
3200:
if it is differentiable with respect to some (or any) coordinate charts defined around
2047:
1596:
1054:
767:
721:
256:
78:
3172:
if it is differentiable with respect to some (or any) coordinate chart defined around
4180:
4168:
3972:
3624:
3495:
3488:
3135:
3134:
at that point. Such a function is necessarily infinitely differentiable, and in fact
2703:
2520:
1560:
778:
1744:
3925:
3915:
3722:
3516:
3466:
3461:
3404:
3392:
3265:
3130:
Any function that is complex-differentiable in a neighborhood of a point is called
2474:
992:
810:
771:
371:
203:
67:
774:
may be continuous, but fails to be differentiable at the location of the anomaly.
4038:
3982:
3804:
3446:
3366:
2066:
1765:
662:(at every point where it is differentiable) as being shown below (in the section
74:
200:
if its derivative is also a continuous function over the domain of the function
4012:
3977:
3967:
3792:
3550:
3376:
2478:
693:
2449:, but again all of the partial derivatives and directional derivatives exist.
696:
function is continuous (i.e. it has no gaps). It is differentiable everywhere
4217:
3957:
3937:
3854:
3533:
979:
is differentiable. However, this function is not continuously differentiable.
782:
1455:
implies that the derivative of any function satisfies the conclusion of the
4043:
3874:
3289:
3270:
3252:
1835:
785:
states that the set of functions that have a derivative at some point is a
777:
Most functions that occur in practice have derivatives at all points or at
70:
4139:
3910:
3819:
3814:
3436:
3414:
2726:
is automatically differentiable at that point, when viewed as a function
806:
Differentiable functions can be locally approximated by linear functions.
47:
39:
4033:
3992:
3987:
3900:
3809:
3717:
3629:
3609:
1839:
984:
786:
55:
3062:, but it is not complex-differentiable at any point because the limit
2118:
However, the existence of the partial derivatives (or even of all the
398:
4028:
3997:
3895:
3739:
3702:
3639:
3593:
3588:
3573:
1701:{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}
361:{\textstyle f^{\prime }(x),f^{\prime \prime }(x),\ldots ,f^{(k)}(x)}
3930:
3762:
1499:
continuously differentiable functions are sometimes said to be of
716:
77:
at each interior point) and does not contain any break, angle, or
4053:
3890:
3844:
3767:
3667:
3662:
3614:
688:
30:
4068:
4048:
3920:
3712:
3869:
3849:
3839:
3834:
3829:
3824:
3787:
3619:
2776:. This is because the complex-differentiability implies that
3127:
does not exist (e.g., it depends on the angle of approach).
586:{\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}
3859:
3018:
is differentiable at every point, viewed as the 2-variable
2428:
2234:
1168:
911:{\displaystyle f(x)=x^{2}\sin \left({\tfrac {1}{x}}\right)}
392:
3253:"Über die Baire'sche Kategorie gewisser Funktionenmengen"
1803:
368:
exist and are continuous over the domain of the function
3141:
2462:
2458:
Smoothness § Multivariate differentiability classes
2084:
If all the partial derivatives of a function exist in a
3148:
Differentiable manifold § Differentiable functions
3068:
2931:
2769:{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}
2706:
2670:
2523:
2487:
1747:
1650:
1619:
1023:
995:
893:
443:
374:
310:
279:
206:
3277:
3120:{\textstyle \lim _{h\to 0}{\frac {h+{\bar {h}}}{2h}}}
3027:
2967:
2785:
2732:
2552:
2264:
2145:
1864:
1776:
1714:
1599:
1571:
1537:
1507:
1474:
1431:
1335:
1303:
1188:
1067:
950:
924:
856:
820:
663:
637:
504:
475:
409:
259:
231:
119:
2106:, then the function is differentiable at that point
3011:{\displaystyle f(z)={\frac {z+{\overline {z}}}{2}}}
706:= 0, where it makes a sharp turn as it crosses the
677:
399:
Differentiability of real functions of one variable
3119:
3054:
3010:
2953:
2914:
2768:
2718:
2692:
2653:
2535:
2509:
2434:
2240:
2026:
1792:
1756:
1733:
1700:
1605:
1584:
1550:
1520:
1490:
1443:
1417:
1318:
1289:
1174:
1045:
1001:
971:
936:
910:
842:
648:
585:
487:
457:
429:
380:
360:
265:
244:
212:
146:
1963:
1957:
93:is an interior point in the domain of a function
4215:
3070:
2787:
2574:
1866:
1210:
526:
1057:, it is possible for the derivative to have an
1418:{\displaystyle f'(x)=2x\sin(1/x)-\cos(1/x)\;,}
843:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
671:
3305:
1708:all exist and are continuous. If derivatives
158:has a non-vertical tangent line at the point
27:Mathematical function whose derivative exists
3319:
2700:, that is complex-differentiable at a point
1997:
1988:
1968:
1888:
797:
2954:{\textstyle f:\mathbb {C} \to \mathbb {C} }
2693:{\textstyle f:\mathbb {C} \to \mathbb {C} }
2510:{\textstyle f:\mathbb {C} \to \mathbb {C} }
3312:
3298:
1411:
1084:
1080:
619:if it is differentiable at every point of
596:exists. This implies that the function is
188:if it is differentiable at every point of
3269:
3184:are differentiable manifolds, a function
2947:
2939:
2811:
2756:
2741:
2686:
2678:
2598:
2503:
2495:
836:
828:
639:
451:
423:
3196:is said to be differentiable at a point
3168:is said to be differentiable at a point
1825:is said to be differentiable at a point
809:
801:
715:
687:
29:
391:For a multivariable function, as shown
66:of a differentiable function has a non-
14:
4216:
3250:
1804:Differentiability in higher dimensions
3293:
3142:Differentiable functions on manifolds
2463:Differentiability in complex analysis
154:exists. In other words, the graph of
4150:Generative adversarial network (GAN)
3160:, a real or complex-valued function
2081:found in single-variable calculus.
2037:If a function is differentiable at
430:{\displaystyle f:U\to \mathbb {R} }
24:
1810:function of several real variables
1782:
1647:
1625:
1029:
623:. In this case, the derivative of
458:{\textstyle U\subset \mathbb {R} }
307:
285:
25:
4245:
3222:Generalizations of the derivative
4188:
4187:
4167:
3282:. Springer-Verlag. Theorem 17.8.
3278:Hewitt, E; Stromberg, K (1963).
2517:is said to be differentiable at
2097:and are continuous at the point
2003:
1992:
1974:
1960:
1952:
1939:
1935:
1926:
1915:
1905:
1901:
1892:
1879:
1871:
1741:exist for all positive integers
678:Differentiability and continuity
664:Differentiability and continuity
184:is said to be differentiable on
4100:Recurrent neural network (RNN)
4090:Differentiable neural computer
3244:
3100:
3077:
3043:
3031:
2977:
2971:
2943:
2899:
2891:
2884:
2877:
2871:
2857:
2851:
2842:
2830:
2823:
2796:
2751:
2682:
2639:
2633:
2624:
2612:
2583:
2567:
2561:
2499:
2422:
2410:
2404:
2392:
2375:
2363:
2357:
2345:
2335:
2309:
2280:
2268:
2161:
2149:
1945:
1930:
1919:
1896:
1875:
1726:
1720:
1695:
1689:
1684:
1678:
1661:
1655:
1636:
1630:
1435:
1408:
1394:
1382:
1368:
1350:
1344:
1262:
1248:
1217:
1203:
1197:
1182:is differentiable at 0, since
1123:
1109:
1077:
1071:
1040:
1034:
960:
954:
866:
860:
832:
574:
568:
559:
547:
533:
519:
513:
419:
355:
349:
344:
338:
321:
315:
296:
290:
141:
128:
13:
1:
4145:Variational autoencoder (VAE)
4105:Long short-term memory (LSTM)
3372:Computational learning theory
3237:
734:is differentiable at a point
649:{\displaystyle \mathbb {R} .}
4125:Convolutional neural network
2997:
1793:{\displaystyle C^{\infty }.}
1061:. For example, the function
781:point. However, a result of
666:). A function is said to be
58:exists at each point in its
7:
4120:Multilayer perceptron (MLP)
3215:
2079:fundamental increment lemma
1046:{\textstyle f^{\prime }(x)}
1013:continuously differentiable
668:continuously differentiable
198:continuously differentiable
10:
4250:
4196:Artificial neural networks
4110:Gated recurrent unit (GRU)
3336:Differentiable programming
3280:Real and abstract analysis
3232:Differentiable programming
3145:
2466:
2451:
1457:intermediate value theorem
982:
764:The converse does not hold
681:
4163:
4077:
4021:
3950:
3883:
3755:
3655:
3648:
3602:
3566:
3529:Artificial neural network
3509:
3385:
3352:Automatic differentiation
3325:
2445:is not differentiable at
2251:is not differentiable at
798:Differentiability classes
672:Differentiability classes
437:, defined on an open set
147:{\displaystyle f'(x_{0})}
34:A differentiable function
3357:Neuromorphic engineering
3320:Differentiable computing
3055:{\displaystyle f(x,y)=x}
1319:{\displaystyle x\neq 0,}
627:is thus a function from
4130:Residual neural network
3546:Artificial Intelligence
3158:differentiable manifold
2120:directional derivatives
1734:{\displaystyle f^{(n)}}
1444:{\displaystyle x\to 0.}
1059:essential discontinuity
937:{\displaystyle x\neq 0}
224:is said to be of class
44:differentiable function
4229:Multivariable calculus
3271:10.4064/sm-3-1-174-179
3227:Semi-differentiability
3121:
3056:
3012:
2955:
2916:
2770:
2720:
2694:
2655:
2537:
2511:
2454:Multivariable calculus
2436:
2242:
2028:
1794:
1758:
1735:
1702:
1607:
1586:
1552:
1522:
1492:
1491:{\displaystyle C^{0},}
1445:
1425:which has no limit as
1419:
1320:
1291:
1176:
1047:
1003:
980:
973:
972:{\displaystyle f(0)=0}
938:
912:
844:
807:
725:
713:
650:
587:
489:
488:{\displaystyle a\in U}
459:
431:
382:
362:
267:
246:
220:. Generally speaking,
214:
148:
62:. In other words, the
35:
4224:Differential calculus
4085:Neural Turing machine
3673:Human image synthesis
3122:
3057:
3013:
2956:
2917:
2771:
2721:
2695:
2656:
2538:
2512:
2437:
2243:
2059:, and the linear map
2029:
1795:
1759:
1736:
1703:
1608:
1587:
1585:{\displaystyle C^{k}}
1553:
1551:{\displaystyle C^{2}}
1523:
1521:{\displaystyle C^{1}}
1493:
1446:
1420:
1327:differentiation rules
1321:
1297:exists. However, for
1292:
1177:
1048:
1004:
974:
939:
913:
845:
813:
805:
719:
691:
651:
588:
490:
460:
432:
383:
363:
268:
247:
245:{\displaystyle C^{k}}
215:
149:
33:
4176:Computer programming
4155:Graph neural network
3730:Text-to-video models
3708:Text-to-image models
3556:Large language model
3541:Scientific computing
3347:Statistical manifold
3342:Information geometry
3066:
3025:
2965:
2929:
2925:However, a function
2783:
2730:
2704:
2668:
2550:
2521:
2485:
2469:Holomorphic function
2262:
2143:
1862:
1774:
1768:or equivalently, of
1745:
1712:
1617:
1597:
1569:
1535:
1505:
1472:
1464:continuous functions
1429:
1333:
1301:
1186:
1065:
1021:
993:
948:
922:
854:
818:
792:Weierstrass function
635:
502:
473:
441:
407:
372:
277:
257:
229:
204:
117:
3522:In-context learning
3362:Pattern recognition
3251:Banach, S. (1931).
2048:partial derivatives
1529:. A function is of
684:Continuous function
4115:Echo state network
4003:Jürgen Schmidhuber
3698:Facial recognition
3693:Speech recognition
3603:Software libraries
3117:
3084:
3052:
3008:
2951:
2912:
2818:
2816:
2766:
2716:
2690:
2651:
2605:
2603:
2533:
2507:
2432:
2427:
2238:
2233:
2046:, then all of the
2024:
1884:
1790:
1754:
1731:
1698:
1603:
1582:
1548:
1518:
1488:
1466:are said to be of
1441:
1415:
1316:
1287:
1224:
1172:
1167:
1055:jump discontinuity
1043:
1017:if the derivative
999:
981:
969:
934:
908:
902:
840:
808:
726:
714:
646:
583:
540:
495:if the derivative
485:
455:
427:
378:
358:
263:
242:
210:
144:
113:if the derivative
36:
4211:
4210:
3973:Stephen Grossberg
3946:
3945:
3115:
3103:
3069:
3006:
3000:
2904:
2791:
2786:
2646:
2578:
2573:
2481:. So, a function
2390:
2343:
2213:
2183:
2016:
1865:
1606:{\displaystyle k}
1561:second derivative
1559:if the first and
1462:Similarly to how
1453:Darboux's theorem
1275:
1209:
1154:
1131:
901:
581:
525:
266:{\displaystyle k}
103:differentiable at
18:Differentiability
16:(Redirected from
4241:
4234:Smooth functions
4201:Machine learning
4191:
4190:
4171:
3926:Action selection
3916:Self-driving car
3723:Stable Diffusion
3688:Speech synthesis
3653:
3652:
3517:Machine learning
3393:Gradient descent
3314:
3307:
3300:
3291:
3290:
3284:
3283:
3275:
3273:
3248:
3126:
3124:
3123:
3118:
3116:
3114:
3106:
3105:
3104:
3096:
3086:
3083:
3061:
3059:
3058:
3053:
3017:
3015:
3014:
3009:
3007:
3002:
3001:
2993:
2984:
2960:
2958:
2957:
2952:
2950:
2942:
2921:
2919:
2918:
2913:
2905:
2903:
2902:
2894:
2888:
2887:
2870:
2826:
2820:
2817:
2815:
2814:
2802:
2775:
2773:
2772:
2767:
2765:
2764:
2759:
2750:
2749:
2744:
2725:
2723:
2722:
2719:{\textstyle x=a}
2717:
2699:
2697:
2696:
2691:
2689:
2681:
2660:
2658:
2657:
2652:
2647:
2642:
2607:
2604:
2602:
2601:
2589:
2560:
2542:
2540:
2539:
2536:{\textstyle x=a}
2534:
2516:
2514:
2513:
2508:
2506:
2498:
2475:complex analysis
2448:
2441:
2439:
2438:
2433:
2431:
2430:
2391:
2388:
2344:
2341:
2334:
2333:
2321:
2320:
2308:
2303:
2302:
2254:
2247:
2245:
2244:
2239:
2237:
2236:
2230:
2229:
2214:
2211:
2200:
2199:
2184:
2181:
2135:
2114:
2105:
2096:
2065:is given by the
2064:
2058:
2045:
2033:
2031:
2030:
2025:
2017:
2015:
2014:
2013:
2012:
2011:
2006:
1995:
1986:
1985:
1984:
1983:
1982:
1977:
1966:
1955:
1944:
1943:
1942:
1929:
1918:
1910:
1909:
1908:
1895:
1886:
1883:
1882:
1874:
1854:
1833:
1824:
1799:
1797:
1796:
1791:
1786:
1785:
1764:the function is
1763:
1761:
1760:
1755:
1740:
1738:
1737:
1732:
1730:
1729:
1707:
1705:
1704:
1699:
1688:
1687:
1654:
1653:
1629:
1628:
1612:
1610:
1609:
1604:
1591:
1589:
1588:
1583:
1581:
1580:
1557:
1555:
1554:
1549:
1547:
1546:
1527:
1525:
1524:
1519:
1517:
1516:
1497:
1495:
1494:
1489:
1484:
1483:
1450:
1448:
1447:
1442:
1424:
1422:
1421:
1416:
1404:
1378:
1343:
1325:
1323:
1322:
1317:
1296:
1294:
1293:
1288:
1280:
1276:
1271:
1258:
1241:
1240:
1230:
1223:
1196:
1181:
1179:
1178:
1173:
1171:
1170:
1155:
1152:
1132:
1129:
1119:
1102:
1101:
1052:
1050:
1049:
1044:
1033:
1032:
1015:
1014:
1008:
1006:
1005:
1000:
978:
976:
975:
970:
943:
941:
940:
935:
917:
915:
914:
909:
907:
903:
894:
881:
880:
849:
847:
846:
841:
839:
831:
772:vertical tangent
761:
748:
742:
733:
711:
705:
655:
653:
652:
647:
642:
630:
626:
622:
618:
610:
603:
592:
590:
589:
584:
582:
577:
542:
539:
512:
494:
492:
491:
486:
465:, is said to be
464:
462:
461:
456:
454:
436:
434:
433:
428:
426:
387:
385:
384:
379:
367:
365:
364:
359:
348:
347:
314:
313:
289:
288:
272:
270:
269:
264:
251:
249:
248:
243:
241:
240:
223:
219:
217:
216:
211:
195:
191:
187:
183:
179:
157:
153:
151:
150:
145:
140:
139:
127:
112:
100:
96:
92:
21:
4249:
4248:
4244:
4243:
4242:
4240:
4239:
4238:
4214:
4213:
4212:
4207:
4159:
4073:
4039:Google DeepMind
4017:
3983:Geoffrey Hinton
3942:
3879:
3805:Project Debater
3751:
3649:Implementations
3644:
3598:
3562:
3505:
3447:Backpropagation
3381:
3367:Tensor calculus
3321:
3318:
3288:
3287:
3249:
3245:
3240:
3218:
3150:
3144:
3107:
3095:
3094:
3087:
3085:
3073:
3067:
3064:
3063:
3026:
3023:
3022:
2992:
2985:
2983:
2966:
2963:
2962:
2946:
2938:
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2926:
2898:
2890:
2889:
2883:
2863:
2822:
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2819:
2810:
2803:
2792:
2790:
2784:
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2780:
2760:
2755:
2754:
2745:
2740:
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2727:
2705:
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2685:
2677:
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2590:
2579:
2577:
2553:
2551:
2548:
2547:
2522:
2519:
2518:
2502:
2494:
2486:
2483:
2482:
2479:complex numbers
2471:
2465:
2460:
2446:
2426:
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2387:
2385:
2379:
2378:
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2338:
2329:
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2316:
2312:
2304:
2298:
2294:
2287:
2286:
2263:
2260:
2259:
2252:
2232:
2231:
2225:
2221:
2210:
2208:
2202:
2201:
2195:
2191:
2180:
2178:
2168:
2167:
2144:
2141:
2140:
2123:
2113:
2107:
2104:
2098:
2095:
2089:
2067:Jacobian matrix
2060:
2057:
2051:
2044:
2038:
2007:
2002:
2001:
2000:
1996:
1991:
1987:
1978:
1973:
1972:
1971:
1967:
1956:
1951:
1938:
1934:
1933:
1925:
1914:
1904:
1900:
1899:
1891:
1887:
1885:
1878:
1870:
1869:
1863:
1860:
1859:
1842:
1832:
1826:
1812:
1806:
1781:
1777:
1775:
1772:
1771:
1757:{\textstyle n,}
1746:
1743:
1742:
1719:
1715:
1713:
1710:
1709:
1677:
1673:
1646:
1642:
1624:
1620:
1618:
1615:
1614:
1598:
1595:
1594:
1576:
1572:
1570:
1567:
1566:
1542:
1538:
1536:
1533:
1532:
1512:
1508:
1506:
1503:
1502:
1479:
1475:
1473:
1470:
1469:
1430:
1427:
1426:
1400:
1374:
1336:
1334:
1331:
1330:
1302:
1299:
1298:
1254:
1236:
1232:
1231:
1229:
1225:
1213:
1189:
1187:
1184:
1183:
1166:
1165:
1151:
1149:
1143:
1142:
1128:
1126:
1115:
1097:
1093:
1086:
1085:
1066:
1063:
1062:
1028:
1024:
1022:
1019:
1018:
1012:
1011:
994:
991:
990:
987:
949:
946:
945:
923:
920:
919:
892:
888:
876:
872:
855:
852:
851:
835:
827:
819:
816:
815:
800:
760:
754:
744:
741:
735:
729:
707:
701:
686:
680:
638:
636:
633:
632:
628:
624:
620:
616:
608:
601:
543:
541:
529:
505:
503:
500:
499:
474:
471:
470:
450:
442:
439:
438:
422:
408:
405:
404:
401:
373:
370:
369:
337:
333:
306:
302:
284:
280:
278:
275:
274:
258:
255:
254:
236:
232:
230:
227:
226:
221:
205:
202:
201:
193:
189:
185:
181:
177:
166:
159:
155:
135:
131:
120:
118:
115:
114:
111:
105:
98:
94:
91:
85:
75:linear function
28:
23:
22:
15:
12:
11:
5:
4247:
4237:
4236:
4231:
4226:
4209:
4208:
4206:
4205:
4204:
4203:
4198:
4185:
4184:
4183:
4178:
4164:
4161:
4160:
4158:
4157:
4152:
4147:
4142:
4137:
4132:
4127:
4122:
4117:
4112:
4107:
4102:
4097:
4092:
4087:
4081:
4079:
4075:
4074:
4072:
4071:
4066:
4061:
4056:
4051:
4046:
4041:
4036:
4031:
4025:
4023:
4019:
4018:
4016:
4015:
4013:Ilya Sutskever
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3978:Demis Hassabis
3975:
3970:
3968:Ian Goodfellow
3965:
3960:
3954:
3952:
3948:
3947:
3944:
3943:
3941:
3940:
3935:
3934:
3933:
3923:
3918:
3913:
3908:
3903:
3898:
3893:
3887:
3885:
3881:
3880:
3878:
3877:
3872:
3867:
3862:
3857:
3852:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3802:
3797:
3796:
3795:
3785:
3780:
3775:
3770:
3765:
3759:
3757:
3753:
3752:
3750:
3749:
3744:
3743:
3742:
3737:
3727:
3726:
3725:
3720:
3715:
3705:
3700:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3659:
3657:
3650:
3646:
3645:
3643:
3642:
3637:
3632:
3627:
3622:
3617:
3612:
3606:
3604:
3600:
3599:
3597:
3596:
3591:
3586:
3581:
3576:
3570:
3568:
3564:
3563:
3561:
3560:
3559:
3558:
3551:Language model
3548:
3543:
3538:
3537:
3536:
3526:
3525:
3524:
3513:
3511:
3507:
3506:
3504:
3503:
3501:Autoregression
3498:
3493:
3492:
3491:
3481:
3479:Regularization
3476:
3475:
3474:
3469:
3464:
3454:
3449:
3444:
3442:Loss functions
3439:
3434:
3429:
3424:
3419:
3418:
3417:
3407:
3402:
3401:
3400:
3389:
3387:
3383:
3382:
3380:
3379:
3377:Inductive bias
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3331:
3329:
3323:
3322:
3317:
3316:
3309:
3302:
3294:
3286:
3285:
3264:(1): 174–179.
3242:
3241:
3239:
3236:
3235:
3234:
3229:
3224:
3217:
3214:
3143:
3140:
3113:
3110:
3102:
3099:
3093:
3090:
3082:
3079:
3076:
3072:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3005:
2999:
2996:
2991:
2988:
2982:
2979:
2976:
2973:
2970:
2949:
2945:
2941:
2937:
2934:
2923:
2922:
2911:
2908:
2901:
2897:
2893:
2886:
2882:
2879:
2876:
2873:
2869:
2866:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2825:
2813:
2809:
2806:
2801:
2798:
2795:
2789:
2763:
2758:
2753:
2748:
2743:
2738:
2735:
2715:
2712:
2709:
2688:
2684:
2680:
2676:
2673:
2662:
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2650:
2645:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2600:
2596:
2593:
2588:
2585:
2582:
2576:
2572:
2569:
2566:
2563:
2559:
2556:
2532:
2529:
2526:
2505:
2501:
2497:
2493:
2490:
2467:Main article:
2464:
2461:
2443:
2442:
2429:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2386:
2384:
2381:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2339:
2337:
2332:
2328:
2324:
2319:
2315:
2311:
2307:
2301:
2297:
2293:
2292:
2290:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2249:
2248:
2235:
2228:
2224:
2220:
2217:
2209:
2207:
2204:
2203:
2198:
2194:
2190:
2187:
2179:
2177:
2174:
2173:
2171:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2111:
2102:
2093:
2055:
2042:
2035:
2034:
2023:
2020:
2010:
2005:
1999:
1994:
1990:
1981:
1976:
1970:
1965:
1962:
1959:
1954:
1950:
1947:
1941:
1937:
1932:
1928:
1924:
1921:
1917:
1913:
1907:
1903:
1898:
1894:
1890:
1881:
1877:
1873:
1868:
1830:
1805:
1802:
1800:
1789:
1784:
1780:
1753:
1750:
1728:
1725:
1722:
1718:
1697:
1694:
1691:
1686:
1683:
1680:
1676:
1672:
1669:
1666:
1663:
1660:
1657:
1652:
1649:
1645:
1641:
1638:
1635:
1632:
1627:
1623:
1602:
1592:
1579:
1575:
1558:
1545:
1541:
1528:
1515:
1511:
1498:
1487:
1482:
1478:
1440:
1437:
1434:
1414:
1410:
1407:
1403:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1377:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1342:
1339:
1315:
1312:
1309:
1306:
1286:
1283:
1279:
1274:
1270:
1267:
1264:
1261:
1257:
1253:
1250:
1247:
1244:
1239:
1235:
1228:
1222:
1219:
1216:
1212:
1208:
1205:
1202:
1199:
1195:
1192:
1169:
1164:
1161:
1158:
1153: if
1150:
1148:
1145:
1144:
1141:
1138:
1135:
1130: if
1127:
1125:
1122:
1118:
1114:
1111:
1108:
1105:
1100:
1096:
1092:
1091:
1089:
1083:
1079:
1076:
1073:
1070:
1042:
1039:
1036:
1031:
1027:
1016:
1009:is said to be
1002:{\textstyle f}
998:
983:Main article:
968:
965:
962:
959:
956:
953:
933:
930:
927:
906:
900:
897:
891:
887:
884:
879:
875:
871:
868:
865:
862:
859:
838:
834:
830:
826:
823:
799:
796:
758:
739:
694:absolute value
679:
676:
645:
641:
613:differentiable
611:is said to be
607:This function
594:
593:
580:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
546:
538:
535:
532:
528:
524:
521:
518:
515:
511:
508:
484:
481:
478:
467:differentiable
453:
449:
446:
425:
421:
418:
415:
412:
400:
397:
381:{\textstyle f}
377:
357:
354:
351:
346:
343:
340:
336:
332:
329:
326:
323:
320:
317:
312:
309:
305:
301:
298:
295:
292:
287:
283:
262:
252:
239:
235:
213:{\textstyle f}
209:
196:is said to be
175:
164:
143:
138:
134:
130:
126:
123:
109:
101:is said to be
89:
50:variable is a
26:
9:
6:
4:
3:
2:
4246:
4235:
4232:
4230:
4227:
4225:
4222:
4221:
4219:
4202:
4199:
4197:
4194:
4193:
4186:
4182:
4179:
4177:
4174:
4173:
4170:
4166:
4165:
4162:
4156:
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4136:
4133:
4131:
4128:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4086:
4083:
4082:
4080:
4078:Architectures
4076:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4027:
4026:
4024:
4022:Organizations
4020:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3958:Yoshua Bengio
3956:
3955:
3953:
3949:
3939:
3938:Robot control
3936:
3932:
3929:
3928:
3927:
3924:
3922:
3919:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3888:
3886:
3882:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3855:Chinchilla AI
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3794:
3791:
3790:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3760:
3758:
3754:
3748:
3745:
3741:
3738:
3736:
3733:
3732:
3731:
3728:
3724:
3721:
3719:
3716:
3714:
3711:
3710:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3660:
3658:
3654:
3651:
3647:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3607:
3605:
3601:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3571:
3569:
3565:
3557:
3554:
3553:
3552:
3549:
3547:
3544:
3542:
3539:
3535:
3534:Deep learning
3532:
3531:
3530:
3527:
3523:
3520:
3519:
3518:
3515:
3514:
3512:
3508:
3502:
3499:
3497:
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3422:Hallucination
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3021:
3020:real function
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2110:
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2018:
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1911:
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1600:
1593:if the first
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898:
895:
889:
885:
882:
877:
873:
869:
863:
857:
824:
821:
814:The function
812:
804:
795:
793:
788:
784:
783:Stefan Banach
780:
775:
773:
769:
765:
757:
752:
749:must also be
747:
738:
732:
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718:
710:
704:
700:at the point
699:
695:
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253:if its first
237:
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61:
57:
53:
49:
45:
41:
32:
19:
4044:Hugging Face
4008:David Silver
3656:Audio–visual
3510:Applications
3489:Augmentation
3334:
3326:
3279:
3276:. Cited by
3261:
3258:Studia Math.
3256:
3246:
3209:
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3201:
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2924:
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2472:
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2117:
2108:
2099:
2090:
2086:neighborhood
2083:
2074:
2070:
2061:
2052:
2039:
2036:
1851:
1847:
1843:
1836:there exists
1827:
1821:
1817:
1813:
1807:
1613:derivatives
1461:
988:
779:almost every
776:
763:
755:
745:
736:
730:
727:
708:
702:
697:
667:
657:
612:
606:
595:
466:
402:
390:
273:derivatives
197:
172:
168:
161:
106:
102:
86:
83:
71:tangent line
43:
37:
4192:Categories
4140:Autoencoder
4095:Transformer
3963:Alex Graves
3911:OpenAI Five
3815:IBM Watsonx
3437:Convolution
3415:Overfitting
3132:holomorphic
2136:defined by
2088:of a point
989:A function
403:A function
40:mathematics
4218:Categories
4181:Technology
4034:EleutherAI
3993:Fei-Fei Li
3988:Yann LeCun
3901:Q-learning
3884:Decisional
3810:IBM Watson
3718:Midjourney
3610:TensorFlow
3457:Activation
3410:Regression
3405:Clustering
3238:References
3146:See also:
2452:See also:
1855:such that
1840:linear map
985:Smoothness
787:meagre set
751:continuous
682:See also:
660:continuous
598:continuous
56:derivative
4064:MIT CSAIL
4029:Anthropic
3998:Andrew Ng
3896:AlphaZero
3740:VideoPoet
3703:AlphaFold
3640:MindSpore
3594:SpiNNaker
3589:Memristor
3496:Diffusion
3472:Rectifier
3452:Batchnorm
3432:Attention
3427:Adversary
3101:¯
3078:→
2998:¯
2944:→
2861:−
2846:−
2808:∈
2797:→
2752:→
2683:→
2628:−
2595:∈
2584:→
2500:→
2361:≠
2189:≠
2050:exist at
1998:‖
1989:‖
1969:‖
1949:−
1923:−
1889:‖
1876:→
1783:∞
1668:…
1651:′
1648:′
1626:′
1436:→
1392:
1386:−
1366:
1308:≠
1273:ε
1266:−
1260:ε
1246:
1234:ε
1218:→
1215:ε
1137:≠
1107:
1030:′
929:≠
886:
833:→
563:−
534:→
480:∈
448:⊂
420:→
328:…
311:′
308:′
286:′
4172:Portals
3931:Auto-GPT
3763:Word2vec
3567:Hardware
3484:Datasets
3386:Concepts
3216:See also
3136:analytic
2868:′
2558:′
2389:if
2342:if
2212:if
2182:if
1341:′
1194:′
510:′
125:′
68:vertical
52:function
4054:Meta AI
3891:AlphaGo
3875:PanGu-Σ
3845:ChatGPT
3820:Granite
3768:Seq2seq
3747:Whisper
3668:WaveNet
3663:AlexNet
3635:Flux.jl
3615:PyTorch
3467:Sigmoid
3462:Softmax
3327:General
3188::
743:, then
97:, then
46:of one
4069:Huawei
4049:OpenAI
3951:People
3921:MuZero
3783:Gemini
3778:Claude
3713:DALL-E
3625:Theano
2447:(0, 0)
2253:(0, 0)
1770:class
1766:smooth
1565:class
1531:class
1501:class
1468:class
1329:imply
712:-axis.
698:except
60:domain
54:whose
4135:Mamba
3906:SARSA
3870:LLaMA
3865:BLOOM
3850:GPT-J
3840:GPT-4
3835:GPT-3
3830:GPT-2
3825:GPT-1
3788:LaMDA
3620:Keras
3176:. If
3156:is a
2543:when
2069:, an
850:with
770:, or
631:into
64:graph
4059:Mila
3860:PaLM
3793:Bard
3773:BERT
3756:Text
3735:Sora
3204:and
3180:and
2456:and
944:and
918:for
768:cusp
722:cusp
692:The
393:here
79:cusp
48:real
42:, a
3800:NMT
3683:OCR
3678:HWR
3630:JAX
3584:VPU
3579:TPU
3574:IPU
3398:SGD
3266:doi
3212:).
3164:on
3152:If
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2788:lim
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883:sin
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