## 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.