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Ðîçâ’ÿçàííÿ. Çã³äíî ç îçíà÷åííÿì çíà÷åííÿ ôóíêö³¿<br />
f(x) = sgn x (äèâ. ï. 6.1.2) ó òî÷ö³ õ = 0 ³ îäíîñòîðîíí³õ<br />
ãðàíèöü â í³é áóäóòü â³äïîâ³äíî òàê³:<br />
f(0) = 0, f(0 + 0) = 1, f(0 − 0) = −1.<br />
Áåçïîñåðåäíüî âèäíî, ùî í³ îäíà ç ð³âíîñòåé âèäó (6.3.1)<br />
íå âèêîíóºòüñÿ. Òàêèì ÷èíîì, òî÷êà õ = 0 º òî÷êîþ ðîçðèâó<br />
ïåðøîãî ðîäó äëÿ ôóíêö³¿ f(x) = sgn x.<br />
Îçíà÷åííÿ 6.3.8. Òî÷êà ðîçðèâó õ 0 ôóíêö³¿ y = f(x) íàçèâàºòüñÿ<br />
òî÷êîþ ðîçðèâó äðóãîãî ðîäó, ÿêùî â ö³é òî÷ö³ õî÷à<br />
á îäíà ç îäíîñòîðîíí³õ ãðàíèöü äîð³âíþº íåñê³í÷åííîñò³ àáî<br />
çîâñ³ì íå ³ñíóº.<br />
Ïðèêëàä 6.3.3. Ïîêàçàòè, ùî äëÿ ôóíêö³é<br />
fx ( )<br />
1<br />
= ,<br />
x<br />
1<br />
fx ( ) = sin òî÷êà õ =0 º òî÷êîþ ðîçðèâó äðóãîãî ðîäó. ijéñíî,<br />
ó â³äïîâ³äíîñò³ äî ïðèêëàä³â 6.2.3 − 6.2.4 ìàºìî,<br />
x<br />
ùî<br />
1 1<br />
lim =∞, ∃ limsin<br />
.<br />
x<br />
x<br />
x→+ 0 x→0<br />
Îòæå, çà îçíà÷åííÿì 6.3.8 òî÷êà õ =0 º òî÷êîþ ðîçðèâó<br />
äðóãîãî ðîäó äëÿ ðîçãëÿäóâàíèõ ôóíêö³é.<br />
6.3.4. Ïîíÿòòÿ ïðî îäíîñòîðîííþ íåïåðåðâí³ñòü<br />
Ó ôîðìóë³ (6.3.1) âñ³ ð³âíîñò³ ìîæóòü íå âèêîíóâàòèñÿ,<br />
àëå ïðè öüîìó îêðåìî ìîæóòü âèêîíóâàòèñÿ òàê³ ð³âíîñò³:<br />
f(x 0 − 0) = f(x 0 ), (6.3.3)<br />
f(x 0 +0)=f(x 0 ). (6.3.4)<br />
Êîëè âèêîíóºòüñÿ óìîâà (6.3.3), òî êàæóòü, ùî ôóíêö³ÿ<br />
y = f(x) íåïåðåðâíà çë³âà â òî÷ö³ õ 0 ∈ (a, b). ßêùî æ âèêîíóºòüñÿ<br />
óìîâà (6.3.4), òî êàæóòü, ùî ôóíêö³ÿ y = f(x) íåïåðåðâíà<br />
ñïðàâà â òî÷ö³ õ 0 ∈ (a, b).<br />
Çàóâàæåííÿ. ßêùî ôóíêö³ÿ y = f(x) âèçíà÷åíà íà ñåãìåíò³<br />
[a, b], òî â òî÷êàõ à ³ b ìîæíà ãîâîðèòè ò³ëüêè ïðî<br />
îäíîñòîðîííþ íåïåðåðâí³ñòü, à ñàìå â òî÷ö³ à ïðî íåïåðåðâí³ñòü<br />
ñïðàâà, â òî÷ö³ b — çë³âà.<br />
Ïðèêëàä 6.3.4. Äîñë³äèòè íà íåïåðåðâí³ñòü ôóíêö³þ<br />
y = E(x) â òî÷ö³ õ =0.<br />
Çàñòîñóºìî îçíà÷åííÿ äëÿ ôóíêö³¿ àíòüº (ðèñ. 6.3). Çã³äíî<br />
ç ãðàô³êîì ö³º¿ ôóíêö³¿ áóäåìî ìàòè E(0) = 0,<br />
E(0 − 0) = −1, E(0 + 0) = 0. Ïðîñòèé àíàë³ç ôîðìóë ïîêàçóº,<br />
ùî ôóíêö³ÿ y = E(x) â òî÷ö³ õ = 0 ìຠðîçðèâ ïåðøîãî ðîäó,<br />
àëå öÿ ôóíêö³ÿ º íåïåðåðâíîþ ñïðàâà â í³é.<br />
6.3.5. Ëîêàëüí³ âëàñòèâîñò³ íåïåðåðâíèõ ôóíêö³é<br />
Äëÿ íåïåðåðâíèõ â òî÷ö³ õ 0 ∈ (a, b) ôóíêö³é ñïðàâåäëèâ³<br />
òàê³ òåîðåìè.<br />
Òåîðåìà 6.3.2. ßêùî ôóíêö³ÿ y = f(x) íåïåðåðâíà â<br />
òî÷ö³ õ 0 ∈ (a, b), òî â äîñòàòíüî ìàëîìó îêîë³ ö³º¿ òî÷êè<br />
ôóíêö³ÿ îáìåæåíà.<br />
Äîâåäåííÿ. Çàñòîñóºìî îçíà÷åííÿ 6.3.3 íåïåðåðâíîñò³<br />
ôóíêö³¿ y = f(x) â òî÷ö³ õ 0 ∈ (a, b). Çã³äíî ç îçíà÷åííÿì â<br />
δ-îêîë³ òî÷êè õ 0 ñïðàâåäëèâà íåð³âí³ñòü fx ( )- fx (<br />
0)<br />
< e,<br />
çâ³äêè âèïëèâຠïîäâ³éíà íåð³âí³ñòü<br />
fx (<br />
0) -e 0. Äàë³ ñêîðèñòóºìîñÿ ë³âîþ ÷àñòèíîþ íåð³âíîñò³<br />
(6.3.5), äå e= (<br />
1<br />
0 ) 0<br />
2 fx > .  ðåçóëüòàò³ îòðèìàºìî íåð³âí³ñòü<br />
fx ( ) > fx (<br />
1<br />
0) > 0, ÿêà çã³äíî ç òåîðåìîþ 6.3.1 áóäå<br />
2<br />
ñïðàâåäëèâîþ â ÿêîìóñü δ-îêîë³ òî÷êè õ 0 . Îòæå, òåîðåìó<br />
äîâåäåíî.<br />
Çàóâàæåííÿ. Çà äîïîìîãîþ òåîðåìè 6.3.3 îá´ðóíòîâó-<br />
ºòüñÿ ìåòîä ³íòåðâàë³â, ÿêèé óñï³øíî âèêîðèñòîâóºòüñÿ äëÿ<br />
ðîçâ’ÿçóâàííÿ íåð³âíîñòåé.<br />
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