448:; there should be no trailing C recurring. There may be a leading sign, and somewhere there will be a tridecimal point to separate the integer part from the fractional part; these should both be ignored in the sequel. These "digits" can be thought of as having the values 0 to 12 respectively; Conway originally used the digits "+", "−" and "." instead of A, B, C, and underlined all of the base-13 "digits" to clearly distinguish them from the usual base-10 digits and symbols.
25:
1069:
263:
The Conway base 13 function was created as part of a "produce" activity: in this case, the challenge was to produce a simple-to-understand function which takes on every real value in every interval, that is, it is an everywhere
254:
In 2018, a much simpler function with the property that every open set is mapped onto the full real line was published by Aksel
Bergfeldt on the mathematics StackExchange. This function is also nowhere continuous.
921:
834:
549:
725:
1129:
1186:
418:
359:) = 0. (Well-formed means that it starts with a + or − symbol, contains exactly one decimal-point symbol, and otherwise contains only the digits 0–9). For example, if a number
305:
in a unique canonical way; such representations use the digits 0–9 plus three additional symbols, say {A, B, C}. For example, the number 54349589 has a base-13 representation
1486:
1371:
969:
644:
1725:
1555:
1519:
1439:
1682:
1618:
958:
1653:
1304:
603:
576:
167:
1272:
245:
216:
1593:
1335:
1243:
1215:
752:
470:
438:
299:
187:
1741:. But as shown above, the Conway base-13 function is unbounded on every interval around every point; therefore it is not continuous anywhere.
1893:
93:
1782:
65:
1306:
The Conway base-13 function shows that the converse is false: it satisfies the intermediate-value property, but is not continuous.
35:
320:. Of course, most numbers will not be intelligible in this way; for example, the number 3629265 has the base-13 representation
72:
839:
757:
475:
312:
If instead of {A, B, C}, we judiciously choose the symbols {+, −, .}, some numbers in base 13 will have representations that
649:
1870:
79:
1075:
1135:
1309:
In fact, the Conway base-13 function satisfies a much stronger intermediate-value property—on every interval (
387:
61:
1845:
1913:
50:
1918:
1729:
The Conway base-13 function is therefore discontinuous everywhere: a real function that is continuous at
1341:. As a result, it satisfies a much stronger discontinuity property— it is discontinuous everywhere.
1064:{\displaystyle f(\mathrm {12345A3C14.159} \dots _{13})=f(\mathrm {A3C14.159} \dots _{13})=3.14159\dots ,}
269:
1923:
316:
like well-formed decimals in base 10: for example, the number 54349589 has a base-13 representation of
130:
1444:
1347:
1344:
From the above follows even more regarding the discontinuity of the function - its graph is dense in
608:
86:
1412:
is negative) to the beginning. By definition of the Conway base-13 function, the resulting string
1528:
46:
1495:
1415:
1687:
1376:
To prove that the Conway base-13 function satisfies this stronger intermediate property, let (
928:
335:. If from some position onward, the representation looks like a well-formed decimal number
1871:"The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond"
1623:
1826:
1807:
Bernardi, Claudio (February 2016). "Graphs of real functions with pathological behaviors".
1572:. By introducing this modification sufficiently far along the tridecimal representation of
1277:
581:
554:
140:
1248:
221:
192:
8:
265:
248:
1830:
1662:
1598:
1575:
42:
1816:
1320:
1228:
1200:
737:
455:
423:
284:
172:
1885:
1758:
134:
126:
1889:
122:
1907:
119:
1197:
According to the intermediate-value theorem, every continuous real function
1745:
24:
1821:
331:
and considers its base-13 representation as a sequence of symbols
728:
441:
302:
16:
Counterexample to the converse of the intermediate value theorem
133:. In other words, it is a function that satisfies a particular
1521:
will have this property. Thus, if we replace the tail end of
1761: – All derivatives have the intermediate value property
1400:, replace the decimal point with C and indicate the sign of
1396:
as follows: starting with the base-10 representation of
1217:
has the intermediate-value property: on every interval (
916:{\displaystyle f(x)=-x_{1}\dots x_{n}.y_{1}y_{2}\dots .}
829:{\displaystyle Bx_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots ,}
452:
If from some point onwards, the tridecimal expansion of
1763:
Pages displaying short descriptions of redirect targets
544:{\displaystyle Ax_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots }
720:{\displaystyle f(x)=x_{1}\dots x_{n}.y_{1}y_{2}\dots }
1690:
1665:
1626:
1601:
1578:
1531:
1498:
1447:
1418:
1350:
1323:
1280:
1251:
1231:
1203:
1138:
1078:
972:
931:
842:
760:
740:
652:
611:
584:
557:
478:
458:
426:
390:
287:
224:
195:
175:
143:
1737:, i.e. it must be bounded on some interval around
1719:
1676:
1647:
1612:
1587:
1549:
1513:
1480:
1433:
1365:
1329:
1298:
1266:
1237:
1209:
1180:
1123:
1063:
952:
915:
828:
746:
719:
638:
597:
570:
543:
464:
432:
412:
293:
239:
210:
181:
161:
1392:be any real number. Create a base-13 encoding of
1124:{\displaystyle f(\mathrm {B1C234} _{13})=-1.234,}
327:Conway's base-13 function takes in a real number
1905:
1181:{\displaystyle f(\mathrm {1C234A567} _{13})=0.}
413:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
630:
612:
51:introducing citations to additional sources
1846:"Is Conway's base-13 function measurable?"
1789:. 2018-09-27. In an answer to the question
734:Similarly, if the tridecimal expansion of
384:The Conway base-13 function is a function
1820:
1353:
406:
398:
1806:
1659:, in every interval we can find a point
41:Relevant discussion may be found on the
420:defined as follows. Write the argument
275:
1906:
440:value as a tridecimal (a "decimal" in
1388:be a point in that interval, and let
1868:
1783:"Open maps which are not continuous"
18:
1595:you can ensure that the new number
1245:passes through every point between
13:
1156:
1150:
1093:
1087:
1027:
1021:
989:
983:
14:
1935:
1843:
1744:The Conway base-13 function maps
118:is a function created by British
1899:from the original on 2016-08-20.
1655:This proves that for any number
1481:{\displaystyle f({\hat {r}})=r.}
1366:{\displaystyle \mathbb {R} ^{2}}
444:) using 13 symbols as "digits":
34:relies largely or entirely on a
23:
1748:the reals in any interval to 0.
1620:will still lie in the interval
1557:the resulting number will have
639:{\displaystyle \{0,\dots ,9\},}
1837:
1800:
1775:
1705:
1694:
1639:
1627:
1538:
1505:
1466:
1460:
1451:
1425:
1404:by prepending either an A (if
1290:
1284:
1261:
1255:
1169:
1142:
1106:
1082:
1046:
1017:
1008:
976:
941:
935:
852:
846:
662:
656:
402:
234:
228:
205:
199:
156:
144:
1:
1768:
1191:
379:
1492:base-13 string that ends in
270:discontinuous at every point
7:
1752:
1733:must be locally bounded at
1550:{\displaystyle {\hat {r}},}
365:8++2.19+0−−7+3.141592653...
135:intermediate-value property
125:as a counterexample to the
10:
1940:
1787:Stack Exchange Mathematics
1514:{\displaystyle {\hat {r}}}
1434:{\displaystyle {\hat {r}}}
258:
189:takes every value between
131:intermediate value theorem
62:"Conway base 13 function"
1890:10.35834/mjms/1418931955
1720:{\displaystyle f(c')=r.}
1408:is positive) or a B (if
137:— on any interval
953:{\displaystyle f(x)=0.}
363:has the representation
333:{0, 1, ..., 9, +, −, .}
116:Conway base 13 function
1914:Functions and mappings
1721:
1678:
1649:
1648:{\displaystyle (a,b).}
1614:
1589:
1551:
1515:
1482:
1441:has the property that
1435:
1384:) be an interval, let
1367:
1331:
1300:
1268:
1239:
1211:
1182:
1125:
1065:
954:
917:
830:
748:
721:
640:
599:
572:
545:
466:
434:
414:
301:can be represented in
295:
241:
212:
183:
163:
1878:Missouri J. Math. Sci
1722:
1679:
1650:
1615:
1590:
1552:
1516:
1483:
1436:
1368:
1332:
1301:
1299:{\displaystyle f(b).}
1269:
1240:
1212:
1183:
1126:
1066:
955:
918:
831:
749:
722:
641:
600:
598:{\displaystyle y_{j}}
573:
571:{\displaystyle x_{i}}
551:where all the digits
546:
467:
446:0, 1, ..., 9, A, B, C
435:
415:
296:
242:
213:
184:
164:
162:{\displaystyle (a,b)}
1688:
1663:
1624:
1599:
1576:
1529:
1496:
1445:
1416:
1348:
1321:
1278:
1267:{\displaystyle f(a)}
1249:
1229:
1201:
1136:
1076:
970:
929:
840:
758:
738:
650:
609:
582:
555:
476:
456:
424:
388:
375:) = +3.141592653....
285:
276:Sketch of definition
240:{\displaystyle f(b)}
222:
211:{\displaystyle f(a)}
193:
173:
141:
47:improve this article
1869:Oman, Greg (2014).
1831:2016arXiv160207555B
266:surjective function
247:— but is not
1919:John Horton Conway
1717:
1677:{\displaystyle c'}
1674:
1645:
1613:{\displaystyle c'}
1610:
1588:{\displaystyle c,}
1585:
1547:
1511:
1478:
1431:
1363:
1327:
1296:
1264:
1235:
1207:
1178:
1121:
1061:
950:
913:
826:
744:
717:
636:
595:
568:
541:
462:
430:
410:
291:
281:Every real number
237:
208:
179:
159:
1924:Special functions
1541:
1508:
1463:
1428:
1339:every real number
1330:{\displaystyle f}
1238:{\displaystyle f}
1210:{\displaystyle f}
747:{\displaystyle x}
465:{\displaystyle x}
433:{\displaystyle x}
294:{\displaystyle x}
182:{\displaystyle f}
112:
111:
97:
1931:
1900:
1898:
1875:
1861:
1860:
1858:
1856:
1841:
1835:
1834:
1824:
1804:
1798:
1797:
1795:
1794:
1779:
1764:
1759:Darboux function
1726:
1724:
1723:
1718:
1704:
1683:
1681:
1680:
1675:
1673:
1654:
1652:
1651:
1646:
1619:
1617:
1616:
1611:
1609:
1594:
1592:
1591:
1586:
1566:
1556:
1554:
1553:
1548:
1543:
1542:
1534:
1520:
1518:
1517:
1512:
1510:
1509:
1501:
1487:
1485:
1484:
1479:
1465:
1464:
1456:
1440:
1438:
1437:
1432:
1430:
1429:
1421:
1372:
1370:
1369:
1364:
1362:
1361:
1356:
1336:
1334:
1333:
1328:
1317:), the function
1305:
1303:
1302:
1297:
1273:
1271:
1270:
1265:
1244:
1242:
1241:
1236:
1225:), the function
1216:
1214:
1213:
1208:
1187:
1185:
1184:
1179:
1168:
1167:
1162:
1130:
1128:
1127:
1122:
1105:
1104:
1099:
1070:
1068:
1067:
1062:
1045:
1044:
1033:
1007:
1006:
995:
959:
957:
956:
951:
922:
920:
919:
914:
906:
905:
896:
895:
883:
882:
870:
869:
835:
833:
832:
827:
819:
818:
809:
808:
796:
795:
783:
782:
773:
772:
753:
751:
750:
745:
726:
724:
723:
718:
713:
712:
703:
702:
690:
689:
677:
676:
645:
643:
642:
637:
604:
602:
601:
596:
594:
593:
577:
575:
574:
569:
567:
566:
550:
548:
547:
542:
537:
536:
527:
526:
514:
513:
501:
500:
491:
490:
471:
469:
468:
463:
447:
439:
437:
436:
431:
419:
417:
416:
411:
409:
401:
366:
334:
323:
319:
308:
300:
298:
297:
292:
246:
244:
243:
238:
217:
215:
214:
209:
188:
186:
185:
180:
168:
166:
165:
160:
107:
104:
98:
96:
55:
27:
19:
1939:
1938:
1934:
1933:
1932:
1930:
1929:
1928:
1904:
1903:
1896:
1873:
1865:
1864:
1854:
1852:
1842:
1838:
1805:
1801:
1792:
1790:
1781:
1780:
1776:
1771:
1762:
1755:
1697:
1689:
1686:
1685:
1666:
1664:
1661:
1660:
1625:
1622:
1621:
1602:
1600:
1597:
1596:
1577:
1574:
1573:
1564:
1533:
1532:
1530:
1527:
1526:
1500:
1499:
1497:
1494:
1493:
1455:
1454:
1446:
1443:
1442:
1420:
1419:
1417:
1414:
1413:
1357:
1352:
1351:
1349:
1346:
1345:
1337:passes through
1322:
1319:
1318:
1279:
1276:
1275:
1250:
1247:
1246:
1230:
1227:
1226:
1202:
1199:
1198:
1194:
1163:
1146:
1145:
1137:
1134:
1133:
1100:
1086:
1085:
1077:
1074:
1073:
1040:
1037:
1020:
1002:
999:
979:
971:
968:
967:
930:
927:
926:
901:
897:
891:
887:
878:
874:
865:
861:
841:
838:
837:
814:
810:
804:
800:
791:
787:
778:
774:
768:
764:
759:
756:
755:
739:
736:
735:
708:
704:
698:
694:
685:
681:
672:
668:
651:
648:
647:
610:
607:
606:
589:
585:
583:
580:
579:
562:
558:
556:
553:
552:
532:
528:
522:
518:
509:
505:
496:
492:
486:
482:
477:
474:
473:
472:is of the form
457:
454:
453:
445:
425:
422:
421:
405:
397:
389:
386:
385:
382:
364:
332:
321:
317:
306:
286:
283:
282:
278:
261:
223:
220:
219:
194:
191:
190:
174:
171:
170:
169:, the function
142:
139:
138:
108:
102:
99:
56:
54:
40:
28:
17:
12:
11:
5:
1937:
1927:
1926:
1921:
1916:
1902:
1901:
1884:(2): 134–150.
1863:
1862:
1836:
1809:Soft Computing
1799:
1773:
1772:
1770:
1767:
1766:
1765:
1754:
1751:
1750:
1749:
1742:
1727:
1716:
1713:
1710:
1707:
1703:
1700:
1696:
1693:
1672:
1669:
1644:
1641:
1638:
1635:
1632:
1629:
1608:
1605:
1584:
1581:
1546:
1540:
1537:
1507:
1504:
1477:
1474:
1471:
1468:
1462:
1459:
1453:
1450:
1427:
1424:
1374:
1360:
1355:
1342:
1326:
1307:
1295:
1292:
1289:
1286:
1283:
1263:
1260:
1257:
1254:
1234:
1206:
1193:
1190:
1189:
1188:
1177:
1174:
1171:
1166:
1161:
1158:
1155:
1152:
1149:
1144:
1141:
1131:
1120:
1117:
1114:
1111:
1108:
1103:
1098:
1095:
1092:
1089:
1084:
1081:
1071:
1060:
1057:
1054:
1051:
1048:
1043:
1039:
1036:
1032:
1029:
1026:
1023:
1019:
1016:
1013:
1010:
1005:
1001:
998:
994:
991:
988:
985:
982:
978:
975:
961:
960:
949:
946:
943:
940:
937:
934:
923:
912:
909:
904:
900:
894:
890:
886:
881:
877:
873:
868:
864:
860:
857:
854:
851:
848:
845:
825:
822:
817:
813:
807:
803:
799:
794:
790:
786:
781:
777:
771:
767:
763:
743:
732:
716:
711:
707:
701:
697:
693:
688:
684:
680:
675:
671:
667:
664:
661:
658:
655:
635:
632:
629:
626:
623:
620:
617:
614:
592:
588:
565:
561:
540:
535:
531:
525:
521:
517:
512:
508:
504:
499:
495:
489:
485:
481:
461:
429:
408:
404:
400:
396:
393:
381:
378:
377:
376:
325:
310:
290:
277:
274:
268:. It is thus
260:
257:
236:
233:
230:
227:
207:
204:
201:
198:
178:
158:
155:
152:
149:
146:
123:John H. Conway
110:
109:
45:. Please help
31:
29:
22:
15:
9:
6:
4:
3:
2:
1936:
1925:
1922:
1920:
1917:
1915:
1912:
1911:
1909:
1895:
1891:
1887:
1883:
1879:
1872:
1867:
1866:
1851:
1847:
1844:Stein, Noah.
1840:
1832:
1828:
1823:
1818:
1814:
1810:
1803:
1788:
1784:
1778:
1774:
1760:
1757:
1756:
1747:
1743:
1740:
1736:
1732:
1728:
1714:
1711:
1708:
1701:
1698:
1691:
1670:
1667:
1658:
1642:
1636:
1633:
1630:
1606:
1603:
1582:
1579:
1571:
1567:
1560:
1544:
1535:
1524:
1502:
1491:
1475:
1472:
1469:
1457:
1448:
1422:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1375:
1358:
1343:
1340:
1324:
1316:
1312:
1308:
1293:
1287:
1281:
1258:
1252:
1232:
1224:
1220:
1204:
1196:
1195:
1175:
1172:
1164:
1159:
1153:
1147:
1139:
1132:
1118:
1115:
1112:
1109:
1101:
1096:
1090:
1079:
1072:
1058:
1055:
1052:
1049:
1041:
1038:
1034:
1030:
1024:
1014:
1011:
1003:
1000:
996:
992:
986:
980:
973:
966:
965:
964:
963:For example:
947:
944:
938:
932:
924:
910:
907:
902:
898:
892:
888:
884:
879:
875:
871:
866:
862:
858:
855:
849:
843:
823:
820:
815:
811:
805:
801:
797:
792:
788:
784:
779:
775:
769:
765:
761:
741:
733:
730:
714:
709:
705:
699:
695:
691:
686:
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36:single source
32:This article
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1850:mathoverflow
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1908:Categories
1822:1602.07555
1793:2023-07-10
1769:References
1746:almost all
1684:such that
1488:Moreover,
1192:Properties
754:ends with
380:Definition
249:continuous
73:newspapers
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403:→
103:June 2016
43:talk page
1894:Archived
1855:6 August
1753:See also
1702:′
1671:′
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127:converse
1827:Bibcode
1815:: 5–6.
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