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6.3.6. Ãëîáàëüí³ âëàñòèâîñò³ íåïåðåðâíèõ ôóíêö³é,<br />
ÿê³ çàäàí³ íà ñåãìåíò³<br />
Ôóíêö³þ íàçèâàþòü íåïåðåðâíîþ íà ñåãìåíò³ [a, b],<br />
ÿêùî âîíà íåïåðåðâíà â êîæí³é òî÷ö³ ³íòåðâàëó (a, b) ³<br />
íåïåðåðâíà â òî÷ö³ à ñïðàâà ³ â òî÷ö³ b çë³âà.<br />
Äëÿ êðàùîãî ðîçóì³ííÿ ïîíÿòòÿ íåïåðåðâíî¿ ôóíêö³¿ íà<br />
ñåãìåíò³ [a, b] ïîÿñíèìî éîãî ó âèïàäêó ¿¿ ãðàô³÷íîãî çîáðàæåííÿ:<br />
ÿêùî íà åñê³ç ãðàô³êà ôóíêö³¿ y = f(x) ìîæíà íàêëàñòè<br />
îäíó íèòî÷êó, òî ôóíêö³ÿ y = f(x) íà ñåãìåíò³ [a, b]<br />
íåïåðåðâíà.<br />
Íåïåðåðâíà íà ñåãìåíò³ ôóíêö³ÿ ìຠíèçêó âëàñòèâîñòåé.<br />
Ñôîðìóëþºìî ö³ âëàñòèâîñò³ ó âèãëÿä³ òåîðåì.<br />
Òåîðåìà 6.3.4 (Âåéºðøòðàññà ïðî îáìåæåí³ñòü). ßêùî<br />
ôóíêö³ÿ f(x) çàäàíà íà ñåãìåíò³ [a, b] ³ íà öüîìó ñåãìåíò³ º<br />
íåïåðåðâíîþ, òî âîíà íà íüîìó º îáìåæåíîþ.<br />
Ïåðø í³æ ñôîðìóëþâàòè íàñòóïíó òåîðåìó, äàìî òàêå<br />
ïîíÿòòÿ.<br />
Íåõàé íà äåÿê³é ÷èñëîâ³é ìíîæèí³ Õ çàäàíà ôóíêö³ÿ<br />
y = f(x), õ ∈ Õ. Òîä³ êàæóòü, ùî çíà÷åííÿ f(ñ), ñ ∈ Õ º íàéá³ëüøèì<br />
(íàéìåíøèì), ÿêùî äëÿ áóäü-ÿêîãî õ ∈ Õ âèêîíó-<br />
ºòüñÿ íåð³âí³ñòü<br />
fx ( ) £ fc ( ) ( fx ( ) ³ fc ( )).<br />
Òåîðåìà 6.3.5 (Âåéºðøòðàññà ïðî äîñÿãíåííÿ íàéá³ëüøîãî<br />
òà íàéìåíøîãî çíà÷åíü). ßêùî ôóíêö³ÿ f(x) íåïåðåðâíà<br />
íà ñåãìåíò³ [a, b], òî âîíà íà íüîìó äîñÿãຠñâîãî íàéá³ëüøîãî<br />
é íàéìåíøîãî çíà÷åíü.<br />
Òåîðåìà 6.3.6 (Áîëüöàíî 1 — Êîø³ ïðî êîð³íü íåïåðåðâíî¿<br />
ôóíêö³¿). ßêùî ôóíêö³ÿ f(x) º íåïåðåðâíîþ íà ñåãìåíò³<br />
[a, b] ³ íà ê³íöÿõ ñåãìåíòà ìຠð³çí³ çíà÷åííÿ, ïðîòèëåæí³<br />
çà çíàêîì, òî ³ñíóº, ïðèíàéìí³, îäíà òàêà òî÷êà ñ ∈ (a, b),<br />
ùî f(ñ) =0.<br />
Òåîðåìà 6.3.7 (Áîëüöàíî — Êîø³ ïðî ïðîì³æíå çíà-<br />
÷åííÿ íåïåðåðâíî¿ ôóíêö³¿). ßêùî ôóíêö³ÿ f(x) º íåïåðåðâíà<br />
íà ñåãìåíò³ [a, b], òî äëÿ áóäü-ÿêîãî ÷èñëà Ñ, ùî çíàõî-<br />
1<br />
Áîëüöàíî Áåðíàðä (1781 – 1848) — çíàìåíèòèé ÷åñüêèé ìàòåìàòèê,<br />
ô³ëîñîô ³ ëîã³ê.<br />
äèòüñÿ ì³æ f(à) ³ f(b), ³ñíóº, ïðèíàéìí³, îäíà òàêà òî÷êà<br />
ñ ∈ (a, b), ùî f(ñ) =Ñ.<br />
Òåîðåìà 6.3.8 (ïðî ³ñíóâàííÿ îáåðíåíî¿ ôóíêö³¿).<br />
ßêùî ôóíêö³ÿ ó = f(x) çàäàíà íà ñåãìåíò³ [a, b] ³ íà öüîìó<br />
ñåãìåíò³ º íåïåðåðâíîþ ³ ñòðîãî ìîíîòîííîþ, òî äëÿ ö³º¿<br />
ôóíêö³¿ ³ñíóº îáåðíåíà ôóíêö³ÿ y = ϕ(x), ÿêà íà ñåãìåíò³<br />
[c, d] ([c, d] — îáëàñòü çíà÷åíü ôóíêö³¿ ó = f(x)) º òàêîæ<br />
íåïåðåðâíîþ ³ ñòðîãî ìîíîòîííîþ.<br />
Òåîðåìè 6.3.5 — 6.3.8 ïîäàºìî áåç äîâåäåííÿ. Ñòðîãå<br />
äîâåäåííÿ ¿õ âèõîäèòü çà ðàìêè äàíîãî ïîñ³áíèêà. Ñë³ä<br />
ò³ëüêè â³äçíà÷èòè, ùî íà ³íòó¿òèâíîìó ð³âí³ ñïðàâåäëèâ³ñòü<br />
òåîðåì º äîñèòü çðîçóì³ëîþ. Òàê, íàïðèêëàä, íåñòðîãå äîâåäåííÿ<br />
òåîðåìè ïðî êîð³íü íåïåðåðâíî¿ ôóíêö³¿ áàçóºòüñÿ íà<br />
òîìó, ùî ÿê áè ìè íå ç’ºäíàëè íåïåðåðâíîþ êðèâîþ (êðèâà<br />
— îäíà íèòî÷êà) òî÷êè A(a, f(a)), B(b, f(b)) (f(a) >0,<br />
f(b) < 0 àáî f(a) 0), òî îáîâ’ÿçêîâî íåïåðåðâíà êðèâà,<br />
ÿêà çàäàíà ð³âíÿííÿì ó = f(x), ïåðåòíå ³íòåðâàë (a, b)<br />
ä³éñíî¿ â³ñ³ Îx. ×èòà÷åâ³ ðåêîìåíäóºìî çä³éñíèòè òàê³ ñõåìàòè÷í³<br />
ïîáóäîâè ñàìîñò³éíî.<br />
Íà çàê³í÷åííÿ ï. 6.3.6 â³äçíà÷èìî, ùî âñ³ îñíîâí³<br />
åëåìåíòàðí³ ôóíêö³¿ º íåïåðåðâíèìè ó ñâî¿õ îáëàñòÿõ âèçíà÷åííÿ.<br />
6.3.7. Íåïåðåðâí³ ³ ðîçðèâí³ ôóíêö³¿ â ïðèðîä³<br />
òà åêîíîì³ö³<br />
ßê ìè âæå âïåâíèëèñÿ, â ìàòåìàòèö³ ôóíêö³¿ íà äåÿê³é<br />
ìíîæèí³ ðîçä³ëÿþòüñÿ íà äâà êëàñè: 1) êëàñ íåïåðåðâíèõ<br />
ôóíêö³é íà çàäàí³é ìíîæèí³; 2) êëàñ ôóíêö³é, ÿê³ ìàþòü<br />
ðîçðèâè â òî÷êàõ çàäàíî¿ ìíîæèíè. Âèíèêàº, ïðèðîäíî, ïèòàííÿ:<br />
³ñíóþòü ÷è í³ â ðåàëüíîìó æèòò³ ïðîöåñè, ÿê³ õàðàêòåðèçóþòüñÿ<br />
âêàçàíèìè êëàñàìè? ³äïîâ³äü íà öå çàïèòàííÿ<br />
ïîçèòèâíà. Ó á³ëüøîñò³ âèïàäê³â çì³íà äåÿêî¿ âåëè÷èíè â<br />
çàëåæíîñò³ â³ä äðóãî¿ çä³éñíþºòüñÿ ïîñòóïîâî áåç ñòðèáê³â.<br />
Íàïðèêëàä, òåìïåðàòóðà âîäè â ïðèáåðåæíèõ âîäàõ ×îðíîãî<br />
ìîðÿ â ë³òí³é ïåð³îä ÿê ôóíêö³ÿ ÷àñó çì³íþºòüñÿ ïîñòóïîâî.<br />
Àëå òàêèé ïðîöåñ ìîæå ð³çêî çì³íèòèñÿ ï³ä 䳺þ äåÿêèõ<br />
ôàêòîð³â, òàêèõ ÿê óðàãàí, ï³äâîäíà òå÷³ÿ ³ òàêå ³íøå.<br />
Ðîçãëÿíåìî ùå îäèí ïðèêëàä. ³çüìåìî êóáèê ñâèíöþ,<br />
ÿêèé ìຠîá’ºì 1 äì 3 ïðè 0 0 Ñ ³ ð³âíîì³ðíî áóäåìî éîãî<br />
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