25:
1297:
941:
1352:
1914:
1388:
1185:
1119:
1041:
840:
513:
1573:
1149:
384:
461:
1509:
1605:
1419:
1238:
1001:
233:
324:
1662:
262:
1956:
1207:
973:
1862:
1634:
1076:
1121:
is linear if and only if there exists a point at which it is continuous, in which case it is continuous everywhere. Consequently, every non-linear additive function
1688:
855:
706:
1996:
1976:
1934:
1818:
1798:
1774:
1529:
1478:
1458:
1438:
786:
766:
746:
726:
683:
663:
643:
620:
597:
577:
553:
533:
282:
207:
187:
163:
2004:(also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
1243:
2088:
42:
89:
386:. Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values.
61:
68:
1720:
for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even
1302:
75:
2038:"Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données"
57:
2096:
108:
1883:
1357:
1154:
1088:
1010:
809:
478:
1534:
1124:
329:
1733:
1699:
849:
801:
435:
2112:
789:
46:
2073:
1486:
2122:
2068:
82:
2063:
1578:
1393:
1212:
1151:
is discontinuous at every point of its domain. Nevertheless, the restriction of any additive function
978:
212:
287:
1639:
1724:. Consequently, every linear map is either continuous everywhere or else continuous nowhere. Every
1713:
238:
2085:
1939:
1190:
946:
1745:
35:
1823:
134:
1610:
1703:
1046:
2007:
2001:
1721:
464:
142:
8:
1667:
138:
688:
1981:
1961:
1919:
1803:
1783:
1759:
1514:
1463:
1443:
1423:
771:
751:
731:
711:
668:
648:
628:
605:
582:
562:
538:
518:
429:
421:
415:
267:
192:
172:
148:
2080:
2117:
1725:
623:
398:
389:
More general definitions of this kind of function can be obtained, by replacing the
1877:
1777:
788:
will be nowhere continuous. Functions of this type were originally investigated by
2092:
556:
425:
390:
1292:{\displaystyle f{\big \vert }_{r\mathbb {Q} }:r\,\mathbb {Q} \to \mathbb {R} }
2106:
1865:
936:{\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all }}x,y\in \mathbb {R} .}
1717:
844:
394:
472:
166:
122:
1729:
1709:
1082:
1004:
1481:
24:
468:
1732:
and on every infinite-dimensional normed space, there exists some
2037:
1085:
is additive, not all additive maps are linear. An additive map
16:
Function which is not continuous at any point of its domain
1209:
is continuous; explicitly, this means that for every real
1347:{\displaystyle r\,\mathbb {Q} :=\{rq:q\in \mathbb {Q} \}}
1390:
is a non-linear additive function then for every point
1984:
1964:
1942:
1922:
1886:
1826:
1806:
1786:
1762:
1670:
1642:
1613:
1581:
1537:
1517:
1489:
1466:
1446:
1426:
1396:
1360:
1305:
1246:
1215:
1193:
1157:
1127:
1091:
1049:
1013:
981:
949:
858:
812:
774:
754:
734:
714:
691:
671:
651:
631:
608:
585:
565:
541:
521:
481:
438:
332:
290:
270:
241:
215:
195:
175:
151:
1187:
to any real scalar multiple of the rational numbers
708:
then the real-valued function which takes the value
49:. Unsourced material may be challenged and removed.
1990:
1970:
1950:
1928:
1908:
1856:
1812:
1792:
1768:
1682:
1656:
1628:
1599:
1567:
1523:
1503:
1472:
1452:
1432:
1413:
1382:
1346:
1291:
1232:
1201:
1179:
1143:
1113:
1070:
1035:
995:
967:
935:
834:
780:
760:
740:
720:
700:
677:
657:
637:
614:
591:
571:
547:
527:
507:
455:
378:
318:
276:
256:
227:
201:
181:
157:
2035:
795:
2104:
397:, or by using the definition of continuity in a
2042:Journal fĂĽr die reine und angewandte Mathematik
1916:is nowhere continuous, there is a dense subset
1007:and continuous). Furthermore, every linear map
1909:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1383:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1180:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1114:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1036:{\displaystyle L:\mathbb {R} \to \mathbb {R} }
1003:is some constant, is additive (in fact, it is
835:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
508:{\displaystyle \mathbf {1} _{\mathbb {Q} }(x)}
1751:
1568:{\displaystyle f\vert _{D}:D\to \mathbb {R} }
1253:
1693:
1542:
1341:
1318:
1144:{\displaystyle \mathbb {R} \to \mathbb {R} }
379:{\displaystyle |f(x)-f(y)|\geq \varepsilon }
456:{\displaystyle \mathbf {1} _{\mathbb {Q} }}
1880: – even if a real function
2081:Dirichlet Function — from MathWorld
1944:
1902:
1894:
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1404:
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1137:
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1021:
989:
926:
828:
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490:
447:
109:Learn how and when to remove this message
2036:Lejeune Dirichlet, Peter Gustav (1829).
189:is nowhere continuous if for each point
1504:{\displaystyle D\subseteq \mathbb {R} }
2105:
1780:extension has the property that every
420:One example of such a function is the
1776:is nowhere continuous if its natural
409:
47:adding citations to reliable sources
18:
2014:everywhere (inside its domain) and
1354:is a continuous function. Thus if
13:
2097:The Wolfram Demonstrations Project
1739:
1600:{\displaystyle D:=x\,\mathbb {Q} }
1575:is continuous (specifically, take
1414:{\displaystyle x\in \mathbb {R} ,}
1233:{\displaystyle r\in \mathbb {R} ,}
14:
2134:
2056:
1748:is discontinuous at every point.
996:{\displaystyle c\in \mathbb {R} }
228:{\displaystyle \varepsilon >0}
131:everywhere discontinuous function
484:
441:
319:{\displaystyle |x-y|<\delta }
23:
2086:The Modified Dirichlet Function
1734:discontinuous linear functional
1700:Discontinuous linear functional
1657:{\displaystyle D:=\mathbb {Q} }
943:For example, every map of form
907:
34:needs additional citations for
2029:
1898:
1851:
1845:
1836:
1830:
1557:
1372:
1281:
1169:
1133:
1103:
1065:
1059:
1025:
953:
904:
898:
889:
883:
874:
862:
824:
796:Non-trivial additive functions
790:Peter Gustav Lejeune Dirichlet
502:
496:
432:. This function is denoted as
393:by the distance function in a
366:
362:
356:
347:
341:
334:
306:
292:
1:
2022:
2010: – a function
1958:such that the restriction of
1864:is appreciable (that is, not
257:{\displaystyle \delta >0,}
58:"Nowhere continuous function"
1951:{\displaystyle \mathbb {R} }
1202:{\displaystyle \mathbb {Q} }
968:{\displaystyle x\mapsto cx,}
850:Cauchy's functional equation
802:Cauchy's functional equation
7:
2069:Encyclopedia of Mathematics
1871:
1043:is of this form (by taking
404:
127:nowhere continuous function
10:
2139:
1752:Hyperreal characterisation
1697:
1480:is also contained in some
799:
413:
1857:{\displaystyle f(x)-f(y)}
1820:such that the difference
1800:is infinitely close to a
1714:topological vector spaces
1694:Discontinuous linear maps
1629:{\displaystyle x\neq 0,}
1746:Conway base 13 function
1071:{\displaystyle c:=L(1)}
1992:
1972:
1952:
1930:
1910:
1858:
1814:
1794:
1770:
1684:
1658:
1630:
1601:
1569:
1525:
1505:
1474:
1454:
1434:
1415:
1384:
1348:
1293:
1234:
1203:
1181:
1145:
1115:
1072:
1037:
997:
969:
937:
836:
782:
762:
742:
722:
702:
679:
665:and the complement of
659:
639:
616:
593:
573:
549:
529:
509:
457:
380:
320:
278:
258:
229:
203:
183:
169:to real numbers, then
159:
2113:Mathematical analysis
1993:
1973:
1953:
1931:
1911:
1859:
1815:
1795:
1771:
1704:Continuous linear map
1685:
1659:
1631:
1602:
1570:
1526:
1506:
1475:
1455:
1435:
1416:
1385:
1349:
1294:
1235:
1204:
1182:
1146:
1116:
1073:
1038:
998:
970:
938:
837:
783:
768:on the complement of
763:
743:
723:
703:
680:
660:
640:
617:
594:
574:
550:
530:
510:
458:
381:
321:
279:
259:
230:
204:
184:
160:
2064:"Dirichlet-function"
2008:Weierstrass function
1982:
1962:
1940:
1920:
1884:
1824:
1804:
1784:
1760:
1722:uniformly continuous
1668:
1640:
1611:
1579:
1535:
1515:
1487:
1464:
1444:
1440:is discontinuous at
1424:
1394:
1358:
1303:
1244:
1213:
1191:
1155:
1125:
1089:
1047:
1011:
979:
947:
856:
810:
772:
752:
732:
712:
689:
669:
649:
629:
606:
583:
563:
539:
519:
479:
436:
428:, also known as the
330:
288:
268:
264:we can find a point
239:
235:such that for every
213:
193:
173:
149:
141:at any point of its
43:improve this article
1683:{\displaystyle x=0}
910: for all
622:is any subset of a
602:More generally, if
165:is a function from
2123:Types of functions
2091:2019-05-02 at the
1988:
1968:
1948:
1926:
1906:
1854:
1810:
1790:
1766:
1680:
1654:
1626:
1597:
1565:
1521:
1501:
1470:
1450:
1430:
1411:
1380:
1344:
1289:
1230:
1199:
1177:
1141:
1111:
1068:
1033:
993:
965:
933:
832:
778:
758:
738:
718:
701:{\displaystyle X,}
698:
675:
655:
635:
612:
589:
569:
545:
525:
505:
471:both equal to the
453:
430:Dirichlet function
422:indicator function
416:Dirichlet function
410:Dirichlet function
376:
316:
274:
254:
225:
199:
179:
155:
2002:Thomae's function
1991:{\displaystyle D}
1971:{\displaystyle f}
1929:{\displaystyle D}
1813:{\displaystyle y}
1793:{\displaystyle x}
1769:{\displaystyle f}
1726:linear functional
1524:{\displaystyle f}
1473:{\displaystyle x}
1453:{\displaystyle x}
1433:{\displaystyle f}
911:
845:additive function
781:{\displaystyle E}
761:{\displaystyle 0}
741:{\displaystyle E}
721:{\displaystyle 1}
678:{\displaystyle E}
658:{\displaystyle E}
638:{\displaystyle X}
624:topological space
615:{\displaystyle E}
592:{\displaystyle x}
572:{\displaystyle 0}
548:{\displaystyle x}
528:{\displaystyle 1}
475:. By definition,
399:topological space
277:{\displaystyle y}
202:{\displaystyle x}
182:{\displaystyle f}
158:{\displaystyle f}
129:, also called an
119:
118:
111:
93:
2130:
2095:by George Beck,
2077:
2050:
2049:
2033:
1997:
1995:
1994:
1989:
1977:
1975:
1974:
1969:
1957:
1955:
1954:
1949:
1947:
1935:
1933:
1932:
1927:
1915:
1913:
1912:
1907:
1905:
1897:
1878:Blumberg theorem
1863:
1861:
1860:
1855:
1819:
1817:
1816:
1811:
1799:
1797:
1796:
1791:
1775:
1773:
1772:
1767:
1756:A real function
1689:
1687:
1686:
1681:
1663:
1661:
1660:
1655:
1653:
1635:
1633:
1632:
1627:
1606:
1604:
1603:
1598:
1596:
1574:
1572:
1571:
1566:
1564:
1550:
1549:
1530:
1528:
1527:
1522:
1510:
1508:
1507:
1502:
1500:
1479:
1477:
1476:
1471:
1459:
1457:
1456:
1451:
1439:
1437:
1436:
1431:
1420:
1418:
1417:
1412:
1407:
1389:
1387:
1386:
1381:
1379:
1371:
1353:
1351:
1350:
1345:
1340:
1314:
1298:
1296:
1295:
1290:
1288:
1280:
1268:
1267:
1266:
1257:
1256:
1240:the restriction
1239:
1237:
1236:
1231:
1226:
1208:
1206:
1205:
1200:
1198:
1186:
1184:
1183:
1178:
1176:
1168:
1150:
1148:
1147:
1142:
1140:
1132:
1120:
1118:
1117:
1112:
1110:
1102:
1077:
1075:
1074:
1069:
1042:
1040:
1039:
1034:
1032:
1024:
1002:
1000:
999:
994:
992:
974:
972:
971:
966:
942:
940:
939:
934:
929:
912:
909:
848:if it satisfies
841:
839:
838:
833:
831:
823:
787:
785:
784:
779:
767:
765:
764:
759:
747:
745:
744:
739:
727:
725:
724:
719:
707:
705:
704:
699:
684:
682:
681:
676:
664:
662:
661:
656:
644:
642:
641:
636:
621:
619:
618:
613:
598:
596:
595:
590:
578:
576:
575:
570:
554:
552:
551:
546:
534:
532:
531:
526:
514:
512:
511:
506:
495:
494:
493:
487:
462:
460:
459:
454:
452:
451:
450:
444:
426:rational numbers
385:
383:
382:
377:
369:
337:
325:
323:
322:
317:
309:
295:
283:
281:
280:
275:
263:
261:
260:
255:
234:
232:
231:
226:
208:
206:
205:
200:
188:
186:
185:
180:
164:
162:
161:
156:
114:
107:
103:
100:
94:
92:
51:
27:
19:
2138:
2137:
2133:
2132:
2131:
2129:
2128:
2127:
2103:
2102:
2093:Wayback Machine
2062:
2059:
2054:
2053:
2034:
2030:
2025:
1983:
1980:
1979:
1963:
1960:
1959:
1943:
1941:
1938:
1937:
1921:
1918:
1917:
1901:
1893:
1885:
1882:
1881:
1874:
1825:
1822:
1821:
1805:
1802:
1801:
1785:
1782:
1781:
1761:
1758:
1757:
1754:
1742:
1740:Other functions
1706:
1696:
1669:
1666:
1665:
1649:
1641:
1638:
1637:
1612:
1609:
1608:
1592:
1580:
1577:
1576:
1560:
1545:
1541:
1536:
1533:
1532:
1531:'s restriction
1516:
1513:
1512:
1496:
1488:
1485:
1484:
1465:
1462:
1461:
1445:
1442:
1441:
1425:
1422:
1421:
1403:
1395:
1392:
1391:
1375:
1367:
1359:
1356:
1355:
1336:
1310:
1304:
1301:
1300:
1284:
1276:
1262:
1258:
1252:
1251:
1250:
1245:
1242:
1241:
1222:
1214:
1211:
1210:
1194:
1192:
1189:
1188:
1172:
1164:
1156:
1153:
1152:
1136:
1128:
1126:
1123:
1122:
1106:
1098:
1090:
1087:
1086:
1081:Although every
1048:
1045:
1044:
1028:
1020:
1012:
1009:
1008:
988:
980:
977:
976:
948:
945:
944:
925:
908:
857:
854:
853:
827:
819:
811:
808:
807:
804:
798:
773:
770:
769:
753:
750:
749:
733:
730:
729:
713:
710:
709:
690:
687:
686:
670:
667:
666:
650:
647:
646:
645:such that both
630:
627:
626:
607:
604:
603:
584:
581:
580:
564:
561:
560:
557:rational number
540:
537:
536:
520:
517:
516:
489:
488:
483:
482:
480:
477:
476:
446:
445:
440:
439:
437:
434:
433:
418:
412:
407:
365:
333:
331:
328:
327:
305:
291:
289:
286:
285:
269:
266:
265:
240:
237:
236:
214:
211:
210:
194:
191:
190:
174:
171:
170:
150:
147:
146:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
2136:
2126:
2125:
2120:
2115:
2101:
2100:
2083:
2078:
2058:
2057:External links
2055:
2052:
2051:
2027:
2026:
2024:
2021:
2020:
2019:
2016:differentiable
2005:
1999:
1998:is continuous.
1987:
1967:
1946:
1925:
1904:
1900:
1896:
1892:
1889:
1873:
1870:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1809:
1789:
1765:
1753:
1750:
1741:
1738:
1695:
1692:
1679:
1676:
1673:
1652:
1648:
1645:
1625:
1622:
1619:
1616:
1595:
1590:
1587:
1584:
1563:
1559:
1556:
1553:
1548:
1544:
1540:
1520:
1499:
1495:
1492:
1469:
1449:
1429:
1410:
1406:
1402:
1399:
1378:
1374:
1370:
1366:
1363:
1343:
1339:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1313:
1308:
1287:
1283:
1279:
1274:
1271:
1265:
1261:
1255:
1249:
1229:
1225:
1221:
1218:
1197:
1175:
1171:
1167:
1163:
1160:
1139:
1135:
1131:
1109:
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1101:
1097:
1094:
1067:
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1061:
1058:
1055:
1052:
1031:
1027:
1023:
1019:
1016:
991:
987:
984:
964:
961:
958:
955:
952:
932:
928:
924:
921:
918:
915:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
847:
830:
826:
822:
818:
815:
797:
794:
777:
757:
737:
717:
697:
694:
674:
654:
634:
611:
588:
568:
544:
524:
504:
501:
498:
492:
486:
449:
443:
414:Main article:
411:
408:
406:
403:
391:absolute value
375:
372:
368:
364:
361:
358:
355:
352:
349:
346:
343:
340:
336:
315:
312:
308:
304:
301:
298:
294:
273:
253:
250:
247:
244:
224:
221:
218:
209:there is some
198:
178:
154:
117:
116:
99:September 2012
31:
29:
22:
15:
9:
6:
4:
3:
2:
2135:
2124:
2121:
2119:
2116:
2114:
2111:
2110:
2108:
2098:
2094:
2090:
2087:
2084:
2082:
2079:
2075:
2071:
2070:
2065:
2061:
2060:
2047:
2043:
2039:
2032:
2028:
2017:
2013:
2009:
2006:
2003:
2000:
1985:
1965:
1923:
1890:
1887:
1879:
1876:
1875:
1869:
1867:
1866:infinitesimal
1848:
1842:
1839:
1833:
1827:
1807:
1787:
1779:
1763:
1749:
1747:
1737:
1735:
1731:
1727:
1723:
1719:
1718:normed spaces
1715:
1711:
1705:
1701:
1691:
1677:
1674:
1671:
1646:
1643:
1623:
1620:
1617:
1614:
1588:
1585:
1582:
1554:
1551:
1546:
1538:
1518:
1493:
1490:
1483:
1467:
1447:
1427:
1408:
1400:
1397:
1364:
1361:
1333:
1330:
1327:
1324:
1321:
1315:
1306:
1272:
1269:
1259:
1247:
1227:
1219:
1216:
1161:
1158:
1095:
1092:
1084:
1079:
1062:
1056:
1053:
1050:
1017:
1014:
1006:
985:
982:
962:
959:
956:
950:
930:
922:
919:
916:
913:
901:
895:
892:
886:
880:
877:
871:
868:
865:
859:
851:
846:
843:
842:is called an
816:
813:
803:
793:
791:
775:
755:
735:
715:
695:
692:
685:are dense in
672:
652:
632:
625:
609:
600:
586:
566:
558:
542:
522:
499:
474:
470:
466:
431:
427:
423:
417:
402:
400:
396:
392:
387:
373:
370:
359:
353:
350:
344:
338:
313:
310:
302:
299:
296:
271:
251:
248:
245:
242:
222:
219:
216:
196:
176:
168:
152:
144:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2067:
2045:
2041:
2031:
2015:
2011:
1755:
1743:
1712:between two
1707:
1482:dense subset
1080:
805:
601:
599:otherwise.
515:is equal to
473:real numbers
419:
395:metric space
388:
167:real numbers
137:that is not
130:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1299:to the set
806:A function
123:mathematics
2107:Categories
2048:: 157–169.
2023:References
2012:continuous
1730:linear map
1716:, such as
1710:linear map
1698:See also:
1083:linear map
800:See also:
559:and it is
284:such that
139:continuous
69:newspapers
2074:EMS Press
1899:→
1840:−
1778:hyperreal
1636:and take
1618:≠
1558:→
1511:on which
1494:⊆
1401:∈
1373:→
1334:∈
1282:→
1220:∈
1170:→
1134:→
1104:→
1026:→
986:∈
954:↦
923:∈
825:→
374:ε
371:≥
351:−
314:δ
300:−
243:δ
217:ε
2118:Topology
2089:Archived
2018:nowhere.
1872:See also
469:codomain
463:and has
405:Examples
135:function
2076:, 2001
424:of the
133:, is a
83:scholar
1005:linear
975:where
465:domain
143:domain
85:
78:
71:
64:
56:
1728:is a
555:is a
145:. If
90:JSTOR
76:books
1744:The
1702:and
1460:but
1078:).
748:and
467:and
326:and
311:<
246:>
220:>
125:, a
62:news
1978:to
1936:of
1868:).
1690:).
1664:if
1607:if
728:on
579:if
535:if
121:In
45:by
2109::
2072:,
2066:,
2044:.
2040:.
1736:.
1708:A
1647::=
1586::=
1316::=
1054::=
852::
792:.
401:.
2099:.
2046:4
1986:D
1966:f
1945:R
1924:D
1903:R
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1672:x
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1100:R
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1063:1
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1057:L
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960:x
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872:y
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860:f
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817::
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756:0
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716:1
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693:X
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491:Q
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360:y
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249:0
223:0
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177:f
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106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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