Let $f$ be continuous on $(a,b)$ and let $c\in(a,b)$. Suppose $f'(x)$ exists for all $x\in(a,b)\backslash\{c\}$ and $\lim_{x\rightarrow c}f'(x)$ exists. Prove that $f'(c)$ exists.
I don't know what to do on this other than follow definitions. So for all $x\in(a,b)\backslash\{c\}$, the limit $$\lim_{y\rightarrow x}\frac{f(y)-f(x)}{y-x}$$ exists, and moreover, $$\lim_{x\rightarrow c}\lim_{y\rightarrow x}\frac{f(y)-f(x)}{y-x}$$ exists. To prove is that $$\lim_{y\rightarrow c}\frac{f(y)-f(c)}{y-c}$$ exists.
