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A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does
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approaches â1 or 1, the values of this function get larger in magnitude. This function can be made bounded if one restricts its domain to be, for example,
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are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.)
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allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.
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2017:
on that interval. Moreover, continuous functions need not be bounded; for example, the functions
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553:. A real-valued function is bounded if and only if it is bounded from above and below.
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is not a bounded function in the sense of this page's definition (unless
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A Concise
Handbook of Mathematics, Physics, and Engineering Sciences
1981:
in order to be bounded. The set of all bounded functions defined on
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The definition of boundedness can be generalized to functions
1811:, is bounded. More generally, any continuous function from a
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1745:{\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}
1145:
1891:{\displaystyle \sin :\mathbb {C} \rightarrow \mathbb {C} }
1177:{\displaystyle \sin :\mathbb {R} \rightarrow \mathbb {R} }
27:
A mathematical function the set of whose values is bounded
2408:
Ghorpade, Sudhir R.; Limaye, Balmohan V. (2010-03-20).
2329:
Polyanin, Andrei D.; Chernoutsan, Alexei (2010-10-18).
1847:{\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} }
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1443:
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2414:. Springer Science & Business Media. p. 56.
2269:
Mathematics for
Engineers and Scientists, 5th Edition
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are either unbounded or constant as a consequence of
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60:. Unsourced material may be challenged and removed.
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2382:"Liouville theorems - Encyclopedia of Mathematics"
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2053:{\displaystyle g:\mathbb {R} ^{2}\to \mathbb {R} }
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1059:This definition can be extended to any function
2411:A Course in Multivariable Calculus and Analysis
2407:
2447:
771:. The set of all bounded sequences forms the
675:{\displaystyle f=(a_{0},a_{1},a_{2},\ldots )}
2106:{\displaystyle h:(0,1)^{2}\to \mathbb {R} }
2871:Vitale's random BrunnâMinkowski inequality
2454:
2440:
1804:{\displaystyle f:\rightarrow \mathbb {R} }
2099:
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682:is bounded if there exists a real number
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120:Learn how and when to remove this message
2209:{\displaystyle h(x,y):={\frac {1}{x+y}}}
131:
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921:. A family of bounded functions may be
14:
2951:
1347:except for â1 and 1, is unbounded. As
836:taking values in a more general space
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1898:must be unbounded since it is entire.
2884:Applications & related
2403:
2401:
2310:from the original on 2 February 2013
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58:adding citations to reliable sources
29:
2803:Marcinkiewicz interpolation theorem
1320:{\displaystyle f(x)=(x^{2}-1)^{-1}}
24:
2729:Symmetric decreasing rearrangement
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1409:
1385:
991:), but has the weaker property of
912:
786:
529:, then the function is said to be
430:, then the function is said to be
25:
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2254:
1253:{\displaystyle x\in \mathbb {R} }
1494:{\textstyle f(x)=(x^{2}+1)^{-1}}
1084:{\displaystyle f:X\rightarrow Y}
957:{\displaystyle T:X\rightarrow Y}
829:{\displaystyle f:X\rightarrow Y}
34:
2294:"The Sine and Cosine Functions"
2013:is much larger than the set of
1815:into a metric space is bounded.
1225:{\displaystyle |\sin(x)|\leq 1}
1050:{\displaystyle T(M)\subseteq Y}
556:An important special case is a
45:needs additional citations for
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1765:on a closed interval, such as
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1599:inverse trigonometric function
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2699:Convergence almost everywhere
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2247:
1862:. In particular, the complex
1818:All complex-valued functions
1427:{\displaystyle (-\infty ,-2]}
741:{\displaystyle |a_{n}|\leq M}
2266:Jeffrey, Alan (1996-06-13).
1921:which takes the value 0 for
1632:{\displaystyle y=\arctan(x)}
1014:{\displaystyle M\subseteq X}
856:by requiring that the image
597:{\displaystyle \mathbb {N} }
272:{\displaystyle |f(x)|\leq M}
203:if the set of its values is
7:
2866:PrĂ©kopaâLeindler inequality
2719:Locally integrable function
2641:{\displaystyle L^{\infty }}
2220:
2153:{\displaystyle g(x,y):=x+y}
1394:{\displaystyle [2,\infty )}
1138:
1021:are mapped to bounded sets
917:Weaker than boundedness is
794:{\displaystyle l^{\infty }}
10:
2985:
2612:Square-integrable function
1979:does not need to be "nice"
1977:bounded. Thus, a function
482:{\displaystyle f(x)\geq B}
383:{\displaystyle f(x)\leq A}
2883:
2861:MinkowskiâSteiner formula
2831:
2793:
2737:
2686:
2620:
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2533:
2469:
1670:{\displaystyle x=\tan(y)}
1565:{\textstyle |f(x)|\leq 1}
751:for every natural number
2844:Isoperimetric inequality
2849:BrunnâMinkowski theorem
2358:"Extreme Value Theorem"
1601:arctangent defined as:
1501:, defined for all real
1327:, defined for all real
582:is taken to be the set
2704:Convergence in measure
2642:
2386:encyclopediaofmath.org
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993:preserving boundedness
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328:bounded is said to be
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185:
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138:
2818:RieszâFischer theorem
2643:
2602:Polarization identity
2362:mathworld.wolfram.com
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2155:
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1681:for all real numbers
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324:. A function that is
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163:
135:
2823:RieszâThorin theorem
2666:Infimum and supremum
2625:
2551:Lebesgue integration
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2117:
2064:
2021:
2015:continuous functions
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878:{\displaystyle f(X)}
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531:bounded (from) below
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432:bounded (from) above
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238:
215:
175:
152:
54:improve this article
2785:Young's convolution
2724:Measurable function
2607:Pythagorean theorem
2597:Parseval's identity
2546:Integrable function
2356:Weisstein, Eric W.
2242:Uniform boundedness
1860:Liouville's theorem
1763:continuous function
1759:boundedness theorem
984:{\displaystyle T=0}
355:is real-valued and
2969:Types of functions
2906:Probability theory
2808:Plancherel theorem
2714:Integral transform
2661:Chebyshev distance
2638:
2587:Euclidean distance
2520:Minkowski distance
2301:math.dartmouth.edu
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2150:
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2003:
1971:Dirichlet function
1955:
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571:
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420:
400:
380:
345:
314:
294:
269:
221:
207:. In other words,
181:
158:
139:
69:"Bounded function"
2946:
2945:
2879:
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2694:Almost everywhere
2479: &
2421:978-1-4419-1621-1
2342:978-1-4398-0640-1
2279:978-0-412-62150-5
2237:Local boundedness
2204:
1967:irrational number
1958:{\displaystyle x}
1934:{\displaystyle x}
1914:{\displaystyle f}
1740:
1721:
1703:and bounded with
1695:{\displaystyle x}
1586:{\displaystyle x}
1515:{\displaystyle x}
1361:{\displaystyle x}
1340:{\displaystyle x}
1184:is bounded since
1127:{\displaystyle Y}
1105:{\displaystyle X}
923:uniformly bounded
919:local boundedness
902:{\displaystyle Y}
849:{\displaystyle Y}
764:{\displaystyle n}
695:{\displaystyle M}
574:{\displaystyle X}
546:{\displaystyle B}
522:{\displaystyle X}
502:{\displaystyle x}
447:{\displaystyle A}
423:{\displaystyle X}
403:{\displaystyle x}
348:{\displaystyle f}
317:{\displaystyle X}
297:{\displaystyle x}
224:{\displaystyle M}
199:values is called
184:{\displaystyle X}
161:{\displaystyle f}
130:
129:
122:
104:
16:(Redirected from
2976:
2959:Complex analysis
2896:Fourier analysis
2854:Milman's reverse
2837:
2835:Lebesgue measure
2829:
2828:
2813:RiemannâLebesgue
2656:Bounded function
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2556:Taxicab geometry
2511:Measurable space
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2012:
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2006:{\displaystyle }
2004:
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1433:
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1251:
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1228:
1223:
1215:
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1183:
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1125:
1111:
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1103:
1090:
1088:
1087:
1082:
1056:
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1053:
1048:
1020:
1018:
1017:
1012:
990:
988:
987:
982:
963:
961:
960:
955:
930:bounded operator
908:
906:
905:
900:
884:
882:
881:
876:
855:
853:
852:
847:
835:
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832:
827:
800:
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790:
789:
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762:
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739:
731:
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681:
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678:
673:
662:
661:
649:
648:
636:
635:
603:
601:
600:
595:
593:
580:
578:
577:
572:
558:bounded sequence
552:
550:
549:
544:
528:
526:
525:
520:
508:
506:
505:
500:
488:
486:
485:
480:
453:
451:
450:
445:
429:
427:
426:
421:
409:
407:
406:
401:
389:
387:
386:
381:
354:
352:
351:
346:
323:
321:
320:
315:
303:
301:
300:
295:
278:
276:
275:
270:
262:
245:
230:
228:
227:
222:
190:
188:
187:
182:
168:defined on some
167:
165:
164:
159:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
18:Bounded sequence
2984:
2983:
2979:
2978:
2977:
2975:
2974:
2973:
2949:
2948:
2947:
2942:
2875:
2832:
2827:
2789:
2765:HausdorffâYoung
2745:BabenkoâBeckner
2733:
2682:
2632:
2628:
2626:
2623:
2622:
2616:
2560:
2529:
2525:Sequence spaces
2465:
2460:
2430:
2429:
2422:
2406:
2399:
2390:
2388:
2380:
2379:
2375:
2366:
2364:
2354:
2350:
2343:
2327:
2323:
2313:
2311:
2307:
2296:
2292:
2291:
2287:
2280:
2264:
2255:
2250:
2232:Compact support
2223:
2193:
2188:
2165:
2162:
2161:
2118:
2115:
2114:
2098:
2089:
2085:
2065:
2062:
2061:
2045:
2036:
2031:
2030:
2022:
2019:
2018:
1986:
1983:
1982:
1950:
1947:
1946:
1942:rational number
1926:
1923:
1922:
1906:
1903:
1902:
1883:
1875:
1867:
1864:
1863:
1839:
1831:
1823:
1820:
1819:
1796:
1770:
1767:
1766:
1732:
1713:
1708:
1705:
1704:
1687:
1684:
1683:
1644:
1641:
1640:
1606:
1603:
1602:
1578:
1575:
1574:
1551:
1534:
1532:
1529:
1528:
1527:bounded, since
1507:
1504:
1503:
1482:
1478:
1466:
1462:
1442:
1439:
1438:
1401:
1374:
1371:
1370:
1353:
1350:
1349:
1332:
1329:
1328:
1308:
1304:
1292:
1288:
1268:
1265:
1264:
1245:
1237:
1234:
1233:
1211:
1191:
1189:
1186:
1185:
1169:
1161:
1153:
1150:
1149:
1141:
1119:
1116:
1115:
1097:
1094:
1093:
1064:
1061:
1060:
1027:
1024:
1023:
1000:
997:
996:
995:; bounded sets
970:
967:
966:
937:
934:
933:
915:
913:Related notions
894:
891:
890:
861:
858:
857:
841:
838:
837:
809:
806:
805:
785:
781:
779:
776:
775:
756:
753:
752:
727:
721:
717:
712:
710:
707:
706:
687:
684:
683:
657:
653:
644:
640:
631:
627:
616:
613:
612:
606:natural numbers
589:
587:
584:
583:
566:
563:
562:
538:
535:
534:
514:
511:
510:
494:
491:
490:
459:
456:
455:
439:
436:
435:
415:
412:
411:
395:
392:
391:
360:
357:
356:
340:
337:
336:
309:
306:
305:
289:
286:
285:
258:
241:
239:
236:
235:
216:
213:
212:
176:
173:
172:
153:
150:
149:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
2982:
2972:
2971:
2966:
2961:
2944:
2943:
2941:
2940:
2939:
2938:
2933:
2923:
2918:
2913:
2908:
2903:
2898:
2893:
2887:
2885:
2881:
2880:
2877:
2876:
2874:
2873:
2868:
2863:
2858:
2857:
2856:
2846:
2840:
2838:
2826:
2825:
2820:
2815:
2810:
2805:
2799:
2797:
2791:
2790:
2788:
2787:
2782:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2741:
2739:
2735:
2734:
2732:
2731:
2726:
2721:
2716:
2711:
2709:Function space
2706:
2701:
2696:
2690:
2688:
2684:
2683:
2681:
2680:
2675:
2674:
2673:
2663:
2658:
2652:
2650:
2635:
2631:
2618:
2617:
2615:
2614:
2609:
2604:
2599:
2594:
2589:
2584:
2582:CauchyâSchwarz
2579:
2573:
2571:
2562:
2561:
2559:
2558:
2553:
2548:
2542:
2540:
2531:
2530:
2528:
2527:
2522:
2517:
2508:
2503:
2502:
2501:
2491:
2483:
2481:Hilbert spaces
2473:
2471:
2470:Basic concepts
2467:
2466:
2459:
2458:
2451:
2444:
2436:
2428:
2427:
2420:
2397:
2373:
2348:
2341:
2321:
2285:
2278:
2252:
2251:
2249:
2246:
2245:
2244:
2239:
2234:
2229:
2222:
2219:
2218:
2217:
2202:
2199:
2196:
2192:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2101:
2097:
2092:
2088:
2084:
2081:
2078:
2075:
2072:
2069:
2048:
2044:
2039:
2034:
2029:
2026:
2002:
1999:
1996:
1993:
1990:
1954:
1930:
1910:
1899:
1886:
1882:
1878:
1874:
1871:
1842:
1838:
1834:
1830:
1827:
1816:
1799:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1755:
1739:
1736:
1731:
1728:
1725:
1720:
1717:
1712:
1691:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1595:
1582:
1561:
1558:
1554:
1550:
1547:
1544:
1541:
1537:
1511:
1488:
1485:
1481:
1477:
1474:
1469:
1465:
1461:
1458:
1455:
1452:
1449:
1446:
1435:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1390:
1387:
1384:
1381:
1378:
1357:
1336:
1314:
1311:
1307:
1303:
1300:
1295:
1291:
1287:
1284:
1281:
1278:
1275:
1272:
1261:
1248:
1244:
1241:
1221:
1218:
1214:
1210:
1207:
1204:
1201:
1198:
1194:
1172:
1168:
1164:
1160:
1157:
1140:
1137:
1123:
1101:
1080:
1077:
1074:
1071:
1068:
1046:
1043:
1040:
1037:
1034:
1031:
1010:
1007:
1004:
980:
977:
974:
953:
950:
947:
944:
941:
914:
911:
898:
874:
871:
868:
865:
845:
825:
822:
819:
816:
813:
788:
784:
773:sequence space
760:
749:
748:
737:
734:
730:
724:
720:
715:
691:
671:
668:
665:
660:
656:
652:
647:
643:
639:
634:
630:
626:
623:
620:
592:
570:
542:
518:
498:
478:
475:
472:
469:
466:
463:
443:
419:
399:
379:
376:
373:
370:
367:
364:
344:
313:
293:
280:
279:
268:
265:
261:
257:
254:
251:
248:
244:
220:
211:a real number
180:
157:
128:
127:
110:September 2021
42:
40:
33:
26:
9:
6:
4:
3:
2:
2981:
2970:
2967:
2965:
2964:Real analysis
2962:
2960:
2957:
2956:
2954:
2937:
2934:
2932:
2929:
2928:
2927:
2924:
2922:
2921:Sobolev space
2919:
2917:
2916:Real analysis
2914:
2912:
2909:
2907:
2904:
2902:
2901:Lorentz space
2899:
2897:
2894:
2892:
2891:Bochner space
2889:
2888:
2886:
2882:
2872:
2869:
2867:
2864:
2862:
2859:
2855:
2852:
2851:
2850:
2847:
2845:
2842:
2841:
2839:
2836:
2830:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2800:
2798:
2796:
2792:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2742:
2740:
2736:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2691:
2689:
2685:
2679:
2676:
2672:
2669:
2668:
2667:
2664:
2662:
2659:
2657:
2654:
2653:
2651:
2649:
2629:
2619:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2592:Hilbert space
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2574:
2572:
2570:
2568:
2563:
2557:
2554:
2552:
2549:
2547:
2544:
2543:
2541:
2539:
2537:
2532:
2526:
2523:
2521:
2518:
2516:
2512:
2509:
2507:
2506:Measure space
2504:
2500:
2497:
2496:
2495:
2492:
2490:
2488:
2484:
2482:
2478:
2475:
2474:
2472:
2468:
2464:
2457:
2452:
2450:
2445:
2443:
2438:
2437:
2434:
2423:
2417:
2413:
2412:
2404:
2402:
2387:
2383:
2377:
2363:
2359:
2352:
2344:
2338:
2335:. CRC Press.
2334:
2333:
2325:
2306:
2302:
2295:
2289:
2281:
2275:
2272:. CRC Press.
2271:
2270:
2262:
2260:
2258:
2253:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2224:
2200:
2197:
2194:
2190:
2185:
2179:
2176:
2173:
2167:
2147:
2144:
2141:
2138:
2132:
2129:
2126:
2120:
2090:
2082:
2079:
2076:
2070:
2067:
2037:
2027:
2024:
2016:
1997:
1994:
1991:
1980:
1976:
1972:
1968:
1965:
1952:
1943:
1928:
1908:
1901:The function
1900:
1872:
1869:
1861:
1857:
1828:
1825:
1817:
1814:
1813:compact space
1787:
1784:
1781:
1775:
1772:
1764:
1760:
1756:
1754:
1737:
1734:
1729:
1726:
1723:
1718:
1715:
1710:
1702:
1689:
1680:
1661:
1655:
1652:
1649:
1646:
1623:
1617:
1614:
1611:
1608:
1600:
1596:
1593:
1580:
1559:
1556:
1545:
1539:
1526:
1522:
1509:
1486:
1483:
1475:
1472:
1467:
1463:
1456:
1450:
1444:
1437:The function
1436:
1418:
1415:
1412:
1406:
1382:
1379:
1368:
1355:
1334:
1312:
1309:
1301:
1298:
1293:
1289:
1282:
1276:
1270:
1263:The function
1262:
1242:
1239:
1219:
1216:
1205:
1199:
1196:
1158:
1155:
1147:
1143:
1142:
1136:
1134:
1121:
1112:
1099:
1078:
1072:
1069:
1066:
1058:
1044:
1041:
1035:
1029:
1008:
1005:
1002:
994:
978:
975:
972:
964:
951:
945:
942:
939:
931:
926:
924:
920:
910:
896:
888:
869:
863:
843:
823:
817:
814:
811:
802:
782:
774:
758:
735:
732:
722:
718:
705:
704:
703:
689:
666:
663:
658:
654:
650:
645:
641:
637:
632:
628:
621:
618:
611:
607:
581:
568:
559:
554:
540:
532:
516:
496:
476:
473:
467:
461:
441:
433:
417:
397:
377:
374:
368:
362:
342:
333:
331:
327:
311:
291:
284:
266:
263:
252:
246:
234:
233:
232:
218:
210:
206:
202:
198:
194:
178:
171:
155:
148:
144:
134:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: â
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
2738:Inequalities
2678:Uniform norm
2655:
2566:
2535:
2486:
2410:
2389:. Retrieved
2385:
2376:
2365:. Retrieved
2361:
2351:
2331:
2324:
2312:. Retrieved
2300:
2288:
2268:
1974:
1945:
1682:
1573:
1524:
1502:
1348:
1114:
1092:
1022:
992:
932:
927:
916:
803:
750:
561:
557:
555:
530:
431:
334:
329:
325:
281:
209:there exists
200:
140:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
2936:Von Neumann
2750:Chebyshev's
2314:1 September
2227:Bounded set
2113:defined by
887:bounded set
231:such that
143:mathematics
2953:Categories
2931:C*-algebra
2755:Clarkson's
2391:2021-09-01
2367:2021-09-01
2248:References
1944:and 1 for
1854:which are
1679:increasing
702:such that
80:newspapers
2926:*-algebra
2911:Quasinorm
2780:Minkowski
2671:Essential
2634:∞
2463:Lp spaces
2096:→
2043:→
1881:→
1837:→
1794:→
1735:π
1716:π
1711:−
1656:
1618:
1557:≤
1484:−
1416:−
1410:∞
1407:−
1386:∞
1310:−
1299:−
1243:∈
1217:≤
1200:
1167:→
1148:function
1076:→
1042:⊆
1006:⊆
949:→
821:→
787:∞
733:≤
667:…
608:. Thus a
474:≥
375:≤
330:unbounded
264:≤
2775:Markov's
2770:Hölder's
2760:Hanner's
2577:Bessel's
2515:function
2499:Lebesgue
2305:Archived
2221:See also
1761:, every
1572:for all
1232:for all
1139:Examples
610:sequence
560:, where
489:for all
390:for all
147:function
2795:Results
2494:Measure
1757:By the
1753:radians
283:for all
205:bounded
201:bounded
197:complex
94:scholar
2648:spaces
2569:spaces
2538:spaces
2489:spaces
2477:Banach
2418:
2339:
2276:
1856:entire
1615:arctan
96:
89:
82:
75:
67:
2308:(PDF)
2297:(PDF)
1969:(cf.
885:is a
454:. If
191:with
101:JSTOR
87:books
2833:For
2687:Maps
2416:ISBN
2337:ISBN
2316:2021
2274:ISBN
2160:and
2060:and
1730:<
1724:<
1597:The
1146:sine
1144:The
1113:and
193:real
145:, a
137:not.
73:news
1870:sin
1677:is
1653:tan
1639:or
1197:sin
1156:sin
1091:if
925:.
889:in
604:of
533:by
509:in
434:by
410:in
335:If
326:not
304:in
195:or
170:set
141:In
56:by
2955::
2400:^
2384:.
2360:.
2303:.
2299:.
2256:^
2186::=
2139::=
1975:is
1973:)
1525:is
1523:,
928:A
909:.
801:.
332:.
2630:L
2567:L
2536:L
2513:/
2487:L
2455:e
2448:t
2441:v
2424:.
2394:.
2370:.
2345:.
2318:.
2282:.
2201:y
2198:+
2195:x
2191:1
2183:)
2180:y
2177:,
2174:x
2171:(
2168:h
2148:y
2145:+
2142:x
2136:)
2133:y
2130:,
2127:x
2124:(
2121:g
2100:R
2091:2
2087:)
2083:1
2080:,
2077:0
2074:(
2071::
2068:h
2047:R
2038:2
2033:R
2028::
2025:g
2001:]
1998:1
1995:,
1992:0
1989:[
1953:x
1929:x
1909:f
1885:C
1877:C
1873::
1841:C
1833:C
1829::
1826:f
1798:R
1791:]
1788:1
1785:,
1782:0
1779:[
1776::
1773:f
1738:2
1727:y
1719:2
1690:x
1665:)
1662:y
1659:(
1650:=
1647:x
1627:)
1624:x
1621:(
1612:=
1609:y
1594:.
1581:x
1560:1
1553:|
1549:)
1546:x
1543:(
1540:f
1536:|
1510:x
1487:1
1480:)
1476:1
1473:+
1468:2
1464:x
1460:(
1457:=
1454:)
1451:x
1448:(
1445:f
1434:.
1422:]
1419:2
1413:,
1404:(
1389:)
1383:,
1380:2
1377:[
1356:x
1335:x
1313:1
1306:)
1302:1
1294:2
1290:x
1286:(
1283:=
1280:)
1277:x
1274:(
1271:f
1260:.
1247:R
1240:x
1220:1
1213:|
1209:)
1206:x
1203:(
1193:|
1171:R
1163:R
1159::
1122:Y
1100:X
1079:Y
1073:X
1070::
1067:f
1057:.
1045:Y
1039:)
1036:M
1033:(
1030:T
1009:X
1003:M
979:0
976:=
973:T
952:Y
946:X
943::
940:T
897:Y
873:)
870:X
867:(
864:f
844:Y
824:Y
818:X
815::
812:f
783:l
759:n
736:M
729:|
723:n
719:a
714:|
690:M
670:)
664:,
659:2
655:a
651:,
646:1
642:a
638:,
633:0
629:a
625:(
622:=
619:f
591:N
569:X
541:B
517:X
497:x
477:B
471:)
468:x
465:(
462:f
442:A
418:X
398:x
378:A
372:)
369:x
366:(
363:f
343:f
312:X
292:x
267:M
260:|
256:)
253:x
250:(
247:f
243:|
219:M
179:X
156:f
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
