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ïîñë³äîâí³ñòü g(f(x n )). Ïðè öüîìó ìຠì³ñöå ð³âí³ñòü (6.2.11).<br />
Òåîðåìó äîâåäåíî.<br />
Öÿ òåîðåìà äîçâîëÿº îá÷èñëþâàòè ãðàíèö³, ïåðåõîäÿ÷è<br />
â³ä çì³ííî¿ õ äî íîâî¿ çì³ííî¿ y = f(x).<br />
Çàóâàæåííÿ. ßêùî ìຠì³ñöå ð³âí³ñòü<br />
lim gy ( ) = ga ( ), (6.2.12)<br />
y → a<br />
òî ð³âí³ñòü (6.2.11) ìîæíà çàïèñàòè ó âèãëÿä³ ôîðìóëè:<br />
⎛ ⎞<br />
lim gfx (()) = g⎜lim<br />
fx () ⎟<br />
⎝ ⎠ , (6.2.13)<br />
x→x0 x→x0<br />
ç ÿêî¿ âèäíî, ùî çíàê ãðàíèö³ òà çíàê ôóíêö³¿ ïðè âèêîíàíí³<br />
óìîâè (6.2.12) ìîæíà ïåðåñòàâëÿòè.<br />
Ôóíêö³¿, ÿê³ ìàþòü âëàñòèâîñò³ (6.2.11) — (6.2.13), íàçèâàþòüñÿ<br />
íåïåðåðâíèìè â òî÷ö³ x = x 0 , ¿õ ìè ðîçãëÿíåìî â<br />
íàñòóïíîìó ïóíêò³.<br />
6.3. ÍÅÏÅÐÅÐÂͲÑÒÜ ÔÓÍÊÖ²¯<br />
6.3.1. Íåïåðåðâí³ñòü ôóíêö³¿ â òî÷ö³<br />
Íåõàé ôóíêö³ÿ y = f(x) âèçíà÷åíà íà ìíîæèí³<br />
X = ( a, x0) È ( x0,<br />
b)<br />
. Ðàí³øå, ïðè ðîçãëÿä³ ãðàíèö³ ôóíêö³¿ â<br />
òî÷ö³ õ 0 , íàñ íå ö³êàâèëî, ÷è º ôóíêö³ÿ y = f(x) âèçíà÷åíîþ<br />
â òî÷ö³ õ 0 , ÷è í³. Òåïåð ïðèïóñòèìî, ùî ³ â òî÷ö³ õ 0 ôóíêö³ÿ<br />
y = f(x) âèçíà÷åíà, òîáòî ôóíêö³ÿ y = f(x) âèçíà÷åíà íà ³íòåðâàë³<br />
(a, b). Ïðè öüîìó ö³ëêîì ïðèðîäíî ââàæàòè ãðàíèöåþ<br />
ôóíêö³¿ y = f(x) â òî÷ö³ õ 0 ÷èñëî f(x 0 ). Ó çâ’ÿçêó ç öèì<br />
ââîäèòüñÿ ïîíÿòòÿ íåïåðåðâíîñò³ ôóíêö³¿ â òî÷ö³.<br />
Îçíà÷åííÿ 6.3.1. Ôóíêö³ÿ y = f(x) íàçèâàºòüñÿ íåïåðåðâíîþ<br />
â òî÷ö³ õ 0 ∈ (a, b), ÿêùî âèêîíóþòüñÿ òàê³ äâ³ óìîâè:<br />
1) ôóíêö³ÿ y = f(x) âèçíà÷åíà â òî÷ö³ õ 0 ;<br />
2) ³ñíóº ãðàíèöÿ lim fx ( ), ³ âîíà äîð³âíþº çíà÷åííþ ôóíêö³¿<br />
â ñàì³é òî÷ö³ õ 0 ,<br />
x→x0<br />
òîáòî<br />
x→x0<br />
= fx0<br />
.<br />
lim fx ( ) ( )<br />
Ö³ëêîì î÷åâèäíî, ùî ÿêùî ôóíêö³ÿ y = f(x) íåïåðåðâíà<br />
â òî÷ö³ õ 0 ∈ (a, b), òî ðîëü ãðàíèö³ â òî÷ö³ õ 0 â³ä³ãðຠ÷èñëî<br />
f(x 0 ). Òîìó, âèêîðèñòàâøè îçíà÷åííÿ ãðàíèö³ ôóíêö³¿ çà<br />
Êîø³ òà Ãåéíå â òî÷ö³ õ 0 , ìîæíà äàòè òàê³ îçíà÷åííÿ.<br />
Îçíà÷åííÿ 6.3.2. Ôóíêö³ÿ y = f(x) íàçèâàºòüñÿ íåïåðåðâíîþ<br />
â òî÷ö³ õ 0 ∈ (a, b), ÿêùî áóäü-ÿêà ïîñë³äîâí³ñòü {x n },<br />
ÿêà íàëåæèòü ³íòåðâàëó (a, b) ³ çá³ãàºòüñÿ äî òî÷êè<br />
õ 0 ∈ (a, b), íàðîäæóº ïîñë³äîâí³ñòü çíà÷åíü ôóíêö³¿ {f(x n )},<br />
ÿêà çá³ãàºòüñÿ äî ÷èñëà f(x 0 ).<br />
Îçíà÷åííÿ 6.3.3. Ôóíêö³ÿ y = f(x) íàçèâàºòüñÿ íåïåðåðâíîþ<br />
â òî÷ö³ õ 0 ∈ (a, b), ÿêùî ∀ε > 0 ∃ δ(ε) > 0, ùî ∀ õ, ÿê³<br />
çàäîâîëüíÿþòü âèìîãó 0 ≤ |x – õ 0 | < δ(ε) ôóíêö³ÿ y = f(x)<br />
çàäîâîëüíÿº íåð³âí³ñòü |f(x) – f(x 0 )| < ε.<br />
Çàóâàæåííÿ. ϳäêðåñëèìî ùå ðàç, ùî â íàâåäåíèõ<br />
îçíà÷åííÿõ ôóíêö³ÿ y = f(x) âèçíà÷åíà íà ³íòåðâàë³ (a, b).<br />
Êð³ì òîãî, â îçíà÷åíí³ 6.3.3 ââàæàºòüñÿ, ùî δ-îê³ë òî÷êè õ 0<br />
íàëåæèòü ³íòåðâàëó (a, b).<br />
Êîðèñòóþ÷èñü ïîíÿòòÿì îäíîñòîðîíí³õ ãðàíèöü, ìîæíà<br />
ââåñòè<br />
Îçíà÷åííÿ 6.3.4. Ôóíêö³ÿ y = f(x) íàçèâàºòüñÿ íåïåðåðâíîþ<br />
â òî÷ö³ õ 0 ∈ (a, b), ÿêùî âèêîíóþòüñÿ òàê³ 5 óìîâ:<br />
1) ôóíêö³ÿ y = f(x) âèçíà÷åíà â òî÷ö³ õ 0 ;<br />
2) ³ñíóº ë³âîñòîðîííÿ ãðàíèöÿ, òîáòî ³ñíóº ÷èñëî f(x 0 −0);<br />
3) ³ñíóº ïðàâîñòîðîííÿ ãðàíèöÿ, òîáòî ³ñíóº ÷èñëî f(x 0 +0);<br />
4) ë³âîñòîðîííÿ é ïðàâîñòîðîííÿ ãðàíèö³ ð³âí³<br />
f(x 0 − 0) = f(x 0 +0);<br />
5) ë³âîñòîðîííÿ é ïðàâîñòîðîííÿ ãðàíèö³ äîð³âíþþòü<br />
çíà÷åííþ ôóíêö³¿ â òî÷ö³ õ 0 , òîáòî<br />
f(x 0 − 0) = f(x 0 +0)=f(x 0 ). (6.3.1)<br />
Íàðåøò³, äàìî ùå îäíå îçíà÷åííÿ íåïåðåðâíîñò³ ôóíêö³¿<br />
â òî÷ö³. ßê ³ ðàí³øå, ïðèïóñòèìî, ùî ôóíêö³ÿ y = f(x) âèçíà÷åíà<br />
íà ³íòåðâàë³ (a, b). Äàë³ â³çüìåìî äâ³ äîâ³ëüí³ òî÷êè<br />
ç öüîãî ³íòåðâàëó õ 0 ³ õ 0 +∆õ, äå ∆õ = õ − õ 0 .<br />
Òîä³ ÷èñëî ∆õ íàçèâàþòü ïðèðîñòîì àðãóìåíòó, à ÷èñëî<br />
∆ó = f(x 0 + ∆x) − f(x 0 ) — ïðèðîñòîì ôóíêö³¿ y = f(x) â òî÷ö³<br />
õ 0 .<br />
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