Fall 2004 #4

Problem Statement: Let f be a continuous function on [a,b] such that f is differentiable at all points of (a,b) except possibly at a single point x_0\in(a,b). If \lim\limits_{x\rightarrow x_0}f'(x) exists show that f'(x_0) exists and that f'(x_0)=\lim\limits_{x\rightarrow x_0}f'(x).

Proof: f'(x)=\lim\limits_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h} by definition. Consider: \lim\limits_{x\rightarrow x_0}f'(x). Applying the definition stated above we get the following:

\lim\limits_{x\rightarrow x_0}f'(x)=\lim\limits_{x\rightarrow x_0}\lim\limits_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}

=\lim\limits_{h\rightarrow 0}\lim\limits_{x\rightarrow x_0}\dfrac{f(x+h)-f(x)}{h}

=\lim\limits_{h\rightarrow 0}\dfrac{f(x_0+h)-f(x_0)}{h}

=f'(x_0)

\Box

Reflection: The only part of this proof that I’m worried about is the switching the limits. I’m 90% sure this is allowed since we’re given that f is continuous.

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