Here's a problem, that I suspect to lack some information without which it cannot be solved. I'm stating the problem as it is in the book, and also mentioning the additional information that we might need. My question is whether or not my suspicion is correct. However, if it can be done without any additional assumption, any help (even in hint form) would be much appreciated. Thank you.
The Problem : Let $f:(a,b) \to \mathbb{R}$ is differentiable except possibly at $c \in (a,b)$. Assume that $\lim_{x \to c} f'(x)$ exists. To show that $f'(c)$ exists and $f'$ is continuous at $c$.
I believe that we need the additional assumption that
$f$ is continuous at $c$.
Here's a counter example. Let $f:(-1,1) \to \mathbb{R}$ be given by $$ f(x)= \left\{ \begin{array}{lr} 0, ~~for~~ x \in (-1,0] \\ 1, ~~for~~ x \in (0,1) \end{array} \right\} $$
Here, $a=-1, b=1, c=0$. Clearly $f$ is differentiable on $(-1,1) \setminus \{0\}$ and $f'(x)=0 ~\forall~x \in (-1,1) \setminus \{0\}$. Hence $\lim_{x \to 0}f'(x)=0$. But $f$ is not continuous, and hence not differentiable at $0,$ contradicting the statement of the problem.
Your suspicion is correct, and your counterexample shows this.