Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Îçíà÷åííÿ 6.3.5 Ôóíêö³ÿ y = f(x) íàçèâàºòüñÿ íåïåðåðâíîþ<br />
â òî÷ö³ õ 0 ∈ (a, b), ÿêùî<br />
lim ∆ y = 0 . (6.3.2)<br />
∆x→0<br />
Íàâåäåí³ òóò ï’ÿòü îçíà÷åíü íåïåðåðâíîñò³ ôóíêö³¿ â<br />
òî÷ö³ º åêâ³âàëåíòíèìè â òîìó ðîçóì³íí³, ùî êîëè ôóíêö³ÿ<br />
y = f(x) íåïåðåðâíà çà ÿêèì-íåáóäü îçíà÷åííÿì, òî âîíà íåïåðåðâíà<br />
é çà ðåøòîþ îçíà÷åíü, òà íàâïàêè.<br />
6.3.2. Àðèôìåòè÷í³ îïåðàö³¿ íàä íåïåðåðâíèìè<br />
ôóíêö³ÿìè<br />
Ïðè óòâîðåíí³ íîâèõ ôóíêö³é ³ç ìíîæèíè íåïåðåðâíèõ<br />
åëåìåíòàðíèõ ôóíêö³é çà äîïîìîãîþ àðèôìåòè÷íèõ ä³é<br />
âèíèêຠïèòàííÿ: ÷è áóäóòü âîíè òåæ íåïåðåðâíèìè. Íà öå<br />
çàïèòàííÿ äຠâ³äïîâ³äü òàêà òåîðåìà.<br />
Òåîðåìà 6.3.1. Íåõàé ôóíêö³¿ f(x) i ϕ(x) íåïåðåðâí³ â<br />
òî÷ö³ õ 0 . Òîä³ â ö³é òî÷ö³ áóäóòü íåïåðåðâíèìè òàê³ ôóíêö³¿:<br />
1)–2) fx ( ) ±j ( x)<br />
; 3) fx ( ) ×j ( x)<br />
; 4)<br />
fx ( )<br />
, äå j ( x0<br />
) ¹ 0.<br />
j( x)<br />
Äîâåäåìî òâåðäæåííÿ 4). Äëÿ öüîãî çàñòîñóºìî ïåðøå<br />
îçíà÷åííÿ íåïåðåðâíîñò³ ôóíêö³¿ â òî÷ö³ ³ òâåðäæåííÿ 4<br />
fx ( )<br />
òåîðåìè 5.2.3. Ââåäåìî ôóíêö³þ y ( x)<br />
= . j ( x)<br />
Îñê³ëüêè j ( x0<br />
) ¹ 0, òî áóäåìî ìàòè<br />
fx ( ) fx (<br />
0)<br />
lim x lim<br />
x → x 0 x → x 0 ϕ ( x ) ϕ ( x0<br />
)<br />
ψ ( ) = = = ψ( x0)<br />
.<br />
Îòæå, ãðàíèöÿ ôóíêö³¿ ψ(x) ïðè x → x 0 äîð³âíþº çíà÷åííþ<br />
ôóíêö³¿ â òî÷ö³ õ 0 , à öå îçíà÷àº, ùî ôóíêö³ÿ ψ(x) íåïåðåðâíà<br />
â òî÷ö³ õ 0 . Òâåðäæåííÿ äîâåäåíî.<br />
6.3.3. Òî÷êè ðîçðèâó ôóíêö³¿ òà ¿õ êëàñèô³êàö³ÿ<br />
ßêùî â îçíà÷åííÿõ 6.3.1 − 6.3.5 íå âèêîíóºòüñÿ õî÷à á<br />
îäíà óìîâà, òî òî÷êà õ 0 ∈ (a, b) íàçèâàºòüñÿ òî÷êîþ ðîçðèâó<br />
ôóíêö³¿.  çàëåæíîñò³ â³ä òîãî, ÿê³ óìîâè íå âèêîíóþòüñÿ,<br />
òî÷êè ðîçðèâó ïîä³ëÿþòüñÿ íà óñóâí³, òî÷êè ðîçðèâó ïåðøîãî<br />
ðîäó òà òî÷êè ðîçðèâó äðóãîãî ðîäó.<br />
Îçíà÷åííÿ 6.3.6. Òî÷êà ðîçðèâó õ 0 ôóíêö³¿ y = f(x) íàçèâàºòüñÿ<br />
óñóâíîþ, ÿêùî â òî÷ö³ õ 0 ôóíêö³ÿ y = f(x) íå âèçíà-<br />
÷åíà, àëå â í³é ³ñíóº ãðàíèöÿ ôóíêö³¿.<br />
Îçíà÷åííÿ ñòàíå çðîçóì³ëèì, ï³ñëÿ òîãî ÿê ìè ââåäåìî<br />
òàê çâàíó “äîâèçíà÷åíó” ôóíêö³þ<br />
ì fx ( ), ÿêùî x¹<br />
x0;<br />
% fx ( ) = í<br />
ï<br />
ïlim fx ( ), ÿêùî x=<br />
x0.<br />
ïî x®<br />
x0<br />
Âîíà çà îçíà÷åííÿì 6.3.1 º íåïåðåðâíîþ â òî÷ö³ õ 0 .<br />
sin x<br />
Ïðèêëàä 6.3.1. Òðåáà äîâèçíà÷èòè ôóíêö³þ fx ( ) = â<br />
x<br />
òî÷ö³ õ = 0 òàê, ùîá äîâèçíà÷åíà ôóíêö³ÿ ñï³âïàäàëà ç<br />
ôóíêö³ºþ f(x) íà âñ³é â³ñ³ 0x, îêð³ì òî÷êè õ =0.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Îñê³ëüêè<br />
sin x<br />
lim = 1 ,<br />
x → 0 x<br />
òî øóêàíîþ ôóíêö³ºþ º ôóíêö³ÿ<br />
ìï<br />
sin x<br />
%<br />
, ÿêùî x ¹ 0;<br />
fx ( ) = ï<br />
í x ïï<br />
ïî 1, ÿêùî x = 0.<br />
sin x<br />
Î÷åâèäíî, ùî òî÷êà õ = 0 äëÿ ôóíêö³¿ fx ( ) = º óñóâíîþ<br />
òî÷êîþ ðîçðèâó.<br />
x<br />
Îçíà÷åííÿ 6.3.7. Òî÷êà ðîçðèâó õ 0 ôóíêö³¿ y = f(x) íàçèâàºòüñÿ<br />
òî÷êîþ ðîçðèâó ïåðøîãî ðîäó, ÿêùî â ö³é òî÷ö³ íå<br />
âèêîíóºòüñÿ ïðèíàéìí³ îäíà ç ð³âíîñòåé (6.3.1).<br />
Ïðèêëàä 6.3.2. Ïîêàçàòè, ùî ôóíêö³ÿ f(x) = sgn x â òî÷ö³<br />
õ = 0 ìຠðîçðèâ ïåðøîãî ðîäó.<br />
184 185
Change language