1455:
in the category of sets, where 1 is the one-point set. Because of this, and the adjunction between
Cartesian products and hom in the category of sets (so there is a canonical isomorphism between functions of two variables and functions of one variable valued in functions of another (single) variable,
1540:
919:
1604:
729:
of a function is a measure of the rate of change of function values with respect to change in input values. Because a constant function does not change, its derivative is 0. This is often written:
470:
1453:
770:
1147:
1191:
836:
1459:
1298:
1335:
1249:
1217:
1088:
1733:
1701:
1666:
1639:
1412:
1773:
1753:
1385:
1355:
1553:
850:
1811:
2279:
2043:
226:
436:
2223:
1918:
1869:
181:
1821:
2274:
574:
does not appear on the right side of the function expression and so its value is "vacuously substituted"; namely
54:
1420:
1023:
412:
211:
2284:
2218:
2202:
2238:
1361:
1111:
166:
1860:
Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S. (2005). "1".
1535:{\displaystyle \operatorname {hom} (X\times Y,Z)\cong \operatorname {hom} (X(\operatorname {hom} (Y,Z))}
2083:
1152:
991:
799:
246:
1934:
2036:
732:
334:
2228:
1543:
2167:
1030:
561:
429:
196:
121:
76:
1254:
2197:
2192:
2187:
2065:
1606:
1303:
719:
458:
29:
1889:
1222:
503:
As a real-valued function of a real-valued argument, a constant function has the general form
2177:
2157:
1910:
392:
1196:
1067:
2243:
2162:
2029:
1706:
1674:
1644:
1617:
1390:
973:
949:
549:
377:
339:
151:
62:
8:
2056:
1776:
557:
311:
301:
296:
2233:
2100:
2095:
2021:
1956:
1758:
1738:
1370:
1340:
1042:
676:
422:
306:
281:
1993:
1838:
1045:'s axiomatization of set theory, the Elementary Theory of the Category of Sets (ETCS).
1990:
1914:
1865:
1817:
1547:
1019:
382:
286:
276:
261:
20:
1550:
of sets as tensor product and the one-point set as tensor unit. In the isomorphisms
2131:
2124:
2119:
1885:
1790:
1038:
935:
623:
372:
349:
106:
2253:
2141:
2136:
2088:
2073:
1034:
959:
939:
469:
367:
344:
291:
1862:
Advanced
Mathematical Concepts - Pre-calculus with Applications, Student Edition
2112:
2107:
387:
2268:
1786:
2248:
1669:
91:
2008:
1955:
Leinster, Tom (27 Jun 2011). "An informal introduction to topos theory".
553:
450:
2052:
2013:
1061:
1001:
726:
635:
271:
266:
2182:
1998:
987:
316:
2172:
1012:
977:
931:
397:
136:
66:
1961:
1415:
795:
is a constant function. For example, given the constant function
402:
648:
because it is a polynomial of degree 0, and its general form is
1905:
Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007).
1599:{\displaystyle \lambda :1\times X\cong X\cong X\times 1:\rho }
709:
axis in the plane. Its graph is symmetric with respect to the
1864:(1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 22.
1008:
666:
is nonzero. This function has no intersection point with the
541:
is the specific constant function where the output value is
183:
914:{\displaystyle y'(x)=\left(x\mapsto -{\sqrt {2}}\right)'=0}
1988:
1859:
948:
is both order-preserving and order-reversing, and if the
2051:
461:
whose (output) value is the same for every input value.
1761:
1741:
1709:
1677:
1647:
1620:
1556:
1462:
1423:
1393:
1373:
1343:
1306:
1257:
1225:
1199:
1155:
1114:
1070:
853:
802:
735:
1904:
1894:(3rd ed.). John Wiley & Sons. p. 122.
112:
1767:
1747:
1727:
1695:
1660:
1633:
1598:
1534:
1447:
1406:
1379:
1349:
1329:
1292:
1243:
1211:
1185:
1141:
1082:
913:
830:
764:
694:. It is the (trivial) constant function and every
1614:, the left and right unitors are the projections
108:
2266:
228:
2074:Zero polynomial (degree undefined or β1 or ββ)
1387:is canonically isomorphic to the function set
718:axis, and therefore a constant function is an
2037:
430:
84:
1935:"Zero Derivative implies Constant Function"
2044:
2030:
437:
423:
93:
1974:Herrlich, Horst and Strecker, George E.,
1960:
1448:{\displaystyle \operatorname {hom} (1,X)}
1954:
1029:A constant function factors through the
772:. The converse is also true. Namely, if
725:In the context where it is defined, the
468:
1836:
1816:. Facts on File, New York. p. 94.
1360:As a corollary, the one-point set is a
1041:. This observation is instrumental for
213:
204:
2267:
1809:
1357:is by definition a constant function.
2025:
1989:
1884:
595:, and so on. No matter what value of
473:An example of a constant function is
232:
217:
198:
187:
172:
153:
142:
123:
97:
78:
1064:to the set of constant functions in
679:. On the other hand, the polynomial
168:
159:
1142:{\displaystyle {\tilde {y}}:X\to Y}
925:
608:The graph of the constant function
495:is 4 regardless of the input value
464:
13:
644:, the constant function is called
14:
2296:
1982:
1186:{\displaystyle {\tilde {y}}(x)=y}
846:is the identically zero function
831:{\displaystyle y(x)=-{\sqrt {2}}}
1018:Every constant function between
1000:, which implies that it is also
138:
129:
1841:. Lamar University. p. 224
1793:if and only if it is constant.
765:{\displaystyle (x\mapsto c)'=0}
55:History of the function concept
1948:
1927:
1898:
1878:
1853:
1830:
1803:
1722:
1710:
1690:
1678:
1529:
1526:
1514:
1505:
1499:
1487:
1469:
1442:
1430:
1287:
1276:
1267:
1261:
1235:
1174:
1168:
1162:
1133:
1121:
1074:
972:Every constant function whose
934:, constant functions are both
883:
868:
862:
812:
806:
749:
742:
736:
626:that passes through the point
1:
2234:Horner's method of evaluation
1796:
1108:, there is a unique function
16:Type of mathematical function
2280:Elementary special functions
1978:, Heldermann Verlag (2007).
1735:respectively to the element
1542:) the category of sets is a
1219:. Conversely, if a function
700:is a root. Its graph is the
7:
2239:Polynomial identity testing
1813:Encyclopedia of Mathematics
568:. The independent variable
10:
2301:
1293:{\displaystyle f(x)=f(x')}
992:full transformation monoid
646:non-zero constant function
530:For example, the function
413:List of specific functions
18:
2211:
2150:
2063:
1330:{\displaystyle x,x'\in X}
692:identically zero function
1544:closed monoidal category
1364:in the category of sets.
1244:{\displaystyle f:X\to Y}
675:axis, meaning it has no
601:is input, the output is
560:of this function is the
19:Not to be confused with
2224:Greatest common divisor
550:domain of this function
484:, because the value of
2275:Elementary mathematics
2096:Quadratic function (2)
1837:Dawkins, Paul (2007).
1810:Tanton, James (2005).
1769:
1749:
1729:
1697:
1662:
1635:
1600:
1536:
1449:
1408:
1381:
1351:
1331:
1294:
1245:
1213:
1212:{\displaystyle x\in X}
1187:
1143:
1084:
1083:{\displaystyle X\to Y}
930:For functions between
915:
832:
766:
634:. In the context of a
500:
2079:Constant function (0)
1911:Pearson Prentice Hall
1779:in the one-point set.
1770:
1750:
1730:
1728:{\displaystyle (x,*)}
1698:
1696:{\displaystyle (*,x)}
1663:
1661:{\displaystyle p_{2}}
1636:
1634:{\displaystyle p_{1}}
1601:
1537:
1450:
1409:
1407:{\displaystyle X^{1}}
1382:
1352:
1332:
1295:
1246:
1214:
1188:
1144:
1085:
916:
833:
783:for all real numbers
767:
472:
2285:Polynomial functions
2212:Tools and algorithms
2132:Quintic function (5)
2120:Quartic function (4)
2057:polynomial functions
1759:
1739:
1707:
1675:
1645:
1618:
1554:
1460:
1421:
1391:
1371:
1341:
1304:
1255:
1223:
1197:
1153:
1112:
1068:
851:
800:
733:
2142:Septic equation (7)
2137:Sextic equation (6)
2084:Linear function (1)
2009:"Constant function"
1994:"Constant Function"
2108:Cubic function (3)
2101:Quadratic equation
1991:Weisstein, Eric W.
1765:
1745:
1725:
1693:
1658:
1631:
1596:
1532:
1445:
1404:
1377:
1347:
1327:
1290:
1241:
1209:
1183:
1139:
1080:
1048:For any non-empty
1043:F. William Lawvere
1020:topological spaces
968:must be constant.
911:
840:The derivative of
828:
762:
552:is the set of all
501:
247:Classes/properties
2262:
2261:
2203:Quasi-homogeneous
1886:Young, Cynthia Y.
1839:"College Algebra"
1768:{\displaystyle *}
1748:{\displaystyle x}
1548:Cartesian product
1380:{\displaystyle X}
1350:{\displaystyle f}
1165:
1124:
1096:and each element
980:are the same set
942:; conversely, if
894:
826:
455:constant function
447:
446:
359:Generalizations
21:function constant
2292:
2125:Quartic equation
2046:
2039:
2032:
2023:
2022:
2018:
2004:
2003:
1967:
1966:
1964:
1952:
1946:
1945:
1943:
1941:
1931:
1925:
1924:
1909:(9th ed.).
1902:
1896:
1895:
1882:
1876:
1875:
1857:
1851:
1850:
1848:
1846:
1834:
1828:
1827:
1807:
1791:locally constant
1785:A function on a
1774:
1772:
1771:
1766:
1754:
1752:
1751:
1746:
1734:
1732:
1731:
1726:
1702:
1700:
1699:
1694:
1667:
1665:
1664:
1659:
1657:
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1640:
1638:
1637:
1632:
1630:
1629:
1612:
1605:
1603:
1602:
1597:
1541:
1539:
1538:
1533:
1454:
1452:
1451:
1446:
1413:
1411:
1410:
1405:
1403:
1402:
1386:
1384:
1383:
1378:
1356:
1354:
1353:
1348:
1336:
1334:
1333:
1328:
1320:
1299:
1297:
1296:
1291:
1286:
1250:
1248:
1247:
1242:
1218:
1216:
1215:
1210:
1192:
1190:
1189:
1184:
1167:
1166:
1158:
1148:
1146:
1145:
1140:
1126:
1125:
1117:
1107:
1101:
1095:
1089:
1087:
1086:
1081:
1059:
1053:
1039:category of sets
999:
985:
967:
957:
947:
936:order-preserving
926:Other properties
922:
920:
918:
917:
912:
904:
900:
896:
895:
890:
861:
845:
839:
837:
835:
834:
829:
827:
822:
794:
788:
782:
771:
769:
768:
763:
755:
717:
715:
708:
706:
699:
689:
674:
672:
665:
661:
643:
638:in one variable
633:
617:
604:
600:
594:
587:
580:
573:
567:
547:
540:
529:
527:
516:
498:
494:
483:
465:Basic properties
439:
432:
425:
237:
236:
230:
222:
221:
215:
207:
206:
202:
192:
191:
185:
177:
176:
170:
162:
161:
157:
147:
146:
140:
132:
131:
127:
117:
116:
110:
102:
101:
95:
87:
86:
82:
49:
26:
25:
2300:
2299:
2295:
2294:
2293:
2291:
2290:
2289:
2265:
2264:
2263:
2258:
2207:
2146:
2089:Linear equation
2059:
2050:
2007:
1985:
1976:Category Theory
1971:
1970:
1953:
1949:
1939:
1937:
1933:
1932:
1928:
1921:
1913:. p. 107.
1903:
1899:
1883:
1879:
1872:
1858:
1854:
1844:
1842:
1835:
1831:
1824:
1808:
1804:
1799:
1760:
1757:
1756:
1740:
1737:
1736:
1708:
1705:
1704:
1676:
1673:
1672:
1652:
1648:
1646:
1643:
1642:
1625:
1621:
1619:
1616:
1615:
1608:
1555:
1552:
1551:
1461:
1458:
1457:
1422:
1419:
1418:
1398:
1394:
1392:
1389:
1388:
1372:
1369:
1368:
1342:
1339:
1338:
1313:
1305:
1302:
1301:
1279:
1256:
1253:
1252:
1224:
1221:
1220:
1198:
1195:
1194:
1157:
1156:
1154:
1151:
1150:
1116:
1115:
1113:
1110:
1109:
1103:
1097:
1091:
1069:
1066:
1065:
1055:
1049:
1035:terminal object
995:
981:
963:
953:
943:
940:order-reversing
932:preordered sets
928:
889:
879:
875:
874:
854:
852:
849:
848:
847:
841:
821:
801:
798:
797:
796:
790:
784:
773:
748:
734:
731:
730:
711:
710:
702:
701:
695:
680:
668:
667:
663:
649:
639:
627:
620:horizontal line
609:
602:
596:
589:
582:
575:
569:
565:
542:
531:
519:
518:
504:
496:
485:
474:
467:
443:
407:
368:Binary relation
354:
321:
241:
235:
227:
220:
212:
201:
197:
190:
182:
175:
167:
156:
152:
145:
137:
126:
122:
115:
107:
100:
92:
81:
77:
36:
24:
17:
12:
11:
5:
2298:
2288:
2287:
2282:
2277:
2260:
2259:
2257:
2256:
2251:
2246:
2241:
2236:
2231:
2226:
2221:
2215:
2213:
2209:
2208:
2206:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2154:
2152:
2148:
2147:
2145:
2144:
2139:
2134:
2129:
2128:
2127:
2117:
2116:
2115:
2113:Cubic equation
2105:
2104:
2103:
2093:
2092:
2091:
2081:
2076:
2070:
2068:
2061:
2060:
2049:
2048:
2041:
2034:
2026:
2020:
2019:
2005:
1984:
1983:External links
1981:
1980:
1979:
1969:
1968:
1947:
1926:
1920:978-0131469686
1919:
1897:
1877:
1871:978-0078682278
1870:
1852:
1829:
1822:
1801:
1800:
1798:
1795:
1783:
1782:
1781:
1780:
1775:is the unique
1764:
1744:
1724:
1721:
1718:
1715:
1712:
1692:
1689:
1686:
1683:
1680:
1655:
1651:
1628:
1624:
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1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1531:
1528:
1525:
1522:
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1516:
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1501:
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1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1401:
1397:
1376:
1365:
1346:
1326:
1323:
1319:
1316:
1312:
1309:
1289:
1285:
1282:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1240:
1237:
1234:
1231:
1228:
1208:
1205:
1202:
1182:
1179:
1176:
1173:
1170:
1164:
1161:
1138:
1135:
1132:
1129:
1123:
1120:
1079:
1076:
1073:
1046:
1027:
1016:
1005:
927:
924:
910:
907:
903:
899:
893:
888:
885:
882:
878:
873:
870:
867:
864:
860:
857:
825:
820:
817:
814:
811:
808:
805:
761:
758:
754:
751:
747:
744:
741:
738:
466:
463:
445:
444:
442:
441:
434:
427:
419:
416:
415:
409:
408:
406:
405:
400:
395:
390:
385:
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375:
370:
364:
361:
360:
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224:
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209:
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188:
179:
173:
164:
154:
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119:
113:
104:
98:
89:
79:
73:
70:
69:
58:
57:
51:
50:
33:
32:
15:
9:
6:
4:
3:
2:
2297:
2286:
2283:
2281:
2278:
2276:
2273:
2272:
2270:
2255:
2254:GrΓΆbner basis
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2225:
2222:
2220:
2219:Factorization
2217:
2216:
2214:
2210:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2155:
2153:
2151:By properties
2149:
2143:
2140:
2138:
2135:
2133:
2130:
2126:
2123:
2122:
2121:
2118:
2114:
2111:
2110:
2109:
2106:
2102:
2099:
2098:
2097:
2094:
2090:
2087:
2086:
2085:
2082:
2080:
2077:
2075:
2072:
2071:
2069:
2067:
2062:
2058:
2054:
2047:
2042:
2040:
2035:
2033:
2028:
2027:
2024:
2016:
2015:
2010:
2006:
2001:
2000:
1995:
1992:
1987:
1986:
1977:
1973:
1972:
1963:
1958:
1951:
1936:
1930:
1922:
1916:
1912:
1908:
1901:
1893:
1892:
1887:
1881:
1873:
1867:
1863:
1856:
1840:
1833:
1825:
1823:0-8160-5124-0
1819:
1815:
1814:
1806:
1802:
1794:
1792:
1788:
1787:connected set
1778:
1762:
1742:
1719:
1716:
1713:
1687:
1684:
1681:
1671:
1670:ordered pairs
1653:
1649:
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1611:
1593:
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1399:
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1374:
1366:
1363:
1359:
1358:
1344:
1324:
1321:
1317:
1314:
1310:
1307:
1283:
1280:
1273:
1270:
1264:
1258:
1238:
1232:
1229:
1226:
1206:
1203:
1200:
1180:
1177:
1171:
1159:
1136:
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1127:
1118:
1106:
1100:
1094:
1077:
1071:
1063:
1058:
1052:
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1044:
1040:
1036:
1032:
1031:one-point set
1028:
1025:
1021:
1017:
1014:
1010:
1006:
1003:
998:
993:
989:
984:
979:
975:
971:
970:
969:
966:
961:
956:
951:
946:
941:
937:
933:
923:
908:
905:
901:
897:
891:
886:
880:
876:
871:
865:
858:
855:
844:
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818:
815:
809:
803:
793:
787:
780:
776:
759:
756:
752:
745:
739:
728:
723:
721:
720:even function
714:
705:
698:
693:
687:
683:
678:
671:
660:
656:
652:
647:
642:
637:
631:
625:
621:
616:
612:
606:
599:
592:
585:
578:
572:
563:
559:
555:
551:
545:
538:
534:
526:
522:
515:
511:
507:
492:
488:
481:
477:
471:
462:
460:
456:
452:
440:
435:
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428:
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421:
420:
418:
417:
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404:
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389:
386:
384:
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371:
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366:
365:
363:
362:
358:
357:
351:
348:
346:
343:
341:
338:
336:
333:
332:
330:
329:
326:Constructions
325:
324:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
283:
280:
278:
275:
273:
270:
268:
265:
263:
260:
258:
255:
254:
252:
251:
248:
245:
244:
238:
225:
223:
210:
208:
195:
193:
180:
178:
165:
163:
150:
148:
135:
133:
120:
118:
105:
103:
90:
88:
75:
74:
72:
71:
68:
64:
60:
59:
56:
53:
52:
47:
43:
39:
35:
34:
31:
28:
27:
22:
2249:Discriminant
2168:Multivariate
2078:
2012:
1997:
1975:
1950:
1938:. Retrieved
1929:
1906:
1900:
1890:
1880:
1861:
1855:
1843:. Retrieved
1832:
1812:
1805:
1784:
1609:
1104:
1098:
1092:
1056:
1054:, every set
1050:
1007:It has zero
996:
982:
964:
954:
944:
929:
842:
791:
785:
778:
774:
724:
712:
703:
696:
691:
685:
681:
669:
658:
654:
650:
645:
640:
629:
619:
614:
610:
607:
597:
590:
583:
576:
570:
554:real numbers
543:
536:
532:
524:
520:
513:
509:
505:
502:
490:
486:
479:
475:
454:
448:
393:Higher-order
256:
45:
41:
37:
2198:Homogeneous
2193:Square-free
2188:Irreducible
2053:Polynomials
1940:January 12,
1891:Precalculus
1845:January 12,
1607:natural in
677:root (zero)
451:mathematics
378:Multivalued
340:Composition
335:Restriction
2269:Categories
2158:Univariate
2014:PlanetMath
1797:References
1367:Every set
1251:satisfies
1149:such that
1090:. For any
1062:isomorphic
1024:continuous
1002:idempotent
727:derivative
636:polynomial
586:(β2.7) = 4
312:Surjective
302:Measurable
297:Continuous
272:Polynomial
2244:Resultant
2183:Trinomial
2163:Bivariate
1999:MathWorld
1962:1012.5647
1763:∗
1720:∗
1682:∗
1594:ρ
1585:×
1579:≅
1573:≅
1567:×
1558:λ
1546:with the
1512:
1497:
1491:≅
1476:×
1467:
1428:
1362:generator
1322:∈
1236:→
1204:∈
1163:~
1134:→
1122:~
1075:→
988:left zero
887:−
884:↦
819:−
743:↦
562:singleton
317:Bijective
307:Injective
282:Algebraic
61:Types by
2229:Division
2178:Binomial
2173:Monomial
1907:Calculus
1888:(2021).
1755:, where
1318:′
1300:for all
1284:′
1193:for all
1013:gradient
978:codomain
902:′
859:′
753:′
662:, where
517:or just
459:function
398:Morphism
383:Implicit
287:Analytic
277:Rational
262:Identity
257:Constant
67:codomain
44: (
30:Function
1416:hom set
1037:in the
990:of the
962:, then
960:lattice
789:, then
690:is the
622:in the
593:(Ο) = 4
579:(0) = 4
403:Functor
373:Partial
350:Inverse
2066:degree
1917:
1868:
1820:
1033:, the
974:domain
950:domain
556:. The
548:. The
292:Smooth
267:Linear
63:domain
1957:arXiv
1777:point
1414:, or
1009:slope
986:is a
958:is a
781:) = 0
688:) = 0
624:plane
618:is a
558:image
539:) = 4
482:) = 4
457:is a
388:Space
2055:and
1942:2014
1915:ISBN
1866:ISBN
1847:2014
1818:ISBN
1703:and
1668:the
1641:and
976:and
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628:(0,
564:set
512:) =
453:, a
65:and
2064:By
1789:is
1509:hom
1494:hom
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952:of
566:{4}
546:= 4
449:In
2271::
2011:.
1996:.
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777:β²(
722:.
613:=
605:.
588:,
581:,
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231:β
216:β
203:β
186:β
171:β
158:β
141:β
128:β
111:β
109:πΉ
96:β
94:πΉ
85:πΉ
83:β
40:β¦
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2017:.
2002:.
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707:-
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