아무래도 대가가 쓴 책 내용이 깔끔하겠죠...
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토시하나 안틀리고 그래로 배꼈음을 먼저 알려드립니다.
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꼬릿말에 달아두었던 그 책을요.
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Theorem) If F is continuous at x=a and if F is differentiable on some deleted neighborhood of x=a and if lim F'(x)(x->a) exists then F'(a) exists and F'(a)=lim F'(x)(x->a); that is, F' is continuous at x=a
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pf) Let f(x)=F(x)-F(a) and g(x)=x-a. Then lim f(x)=0 (x->a) and lim g(x)=0 (x->a). Also, lim[f'(x)/g'(x)]=lim F'(x)(x->a 둘다) exists. By L'Hospital's rule, lim[f(x)/g(x)] (x->a) exists and equals
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lim f'(x)/g'(x) = lim F'(x) (x->a)
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Therefore
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lim (F(x)-F(a))/(x-a)=lim F'(x) (x->a)
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and so F'(a)=lim F'(x) (x->a)
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This tells us that F(x) is differentiable at x=a and moreover that F' is continuous at x=a
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