Let $f\colon \Bbb R→\Bbb R$ be continuous, and differentiable at every point $x∈\Bbb R−\{c\}$ for some $c∈\Bbb R$.
If $\lim_{x\to c}f'(x)$ exists, then $f$ is differentiable at $x=c$ with $f'(c)=\lim_{x\to c}f'(x)$.
I want to know how to prove that the sentence above is true.
By mean value theorem, for all $x \neq c$, it exists $a_x \in (x,c)$ such that
$$\frac{f(x)-f(c)}{x-c} = f^\prime(a_x)$$ therefore if you denote $\lim\limits_{x \to c} f^\prime(x) =C$, you can prove that $$\lim\limits_{x \to c} \frac{f(x)-f(c)}{x-c} = C$$ with an $\epsilon$-$\delta$ proof. That allows to get the conclusion that $f^\prime(c)=C$.
