Let $a<c<b, f:(a,b) \to \mathbb{R}$ be continuous. Assume $f$ is differentiable at every point of $(a,b) \backslash \{c\}$ and $f'$ has a limit at $c$. Then which of the following are true?
1)$f$ is differentiable at $c$.
2)$f$ need not be differentiable at $c$.
3)$f$ is differentiable at $c$ but $f'(c)$ is not necessarily $\lim_{x \to c}f'(x)$.
4) $f$ is differentiable at $c$ and $\lim_{x \to c}f'(x)=f'(c)$.
Since $f$ is given to be continuous everywhere and is differentiable at every point except at $c$, I feel option 1) is correct, hence 2) is false.
How to look at 3) and 4)?
Note that corners and cusps are out of the question because either would produce a jump discontinuity (of finite length) in the image of $f'$. The remaining options of both a removable discontinuity or a hyperbolic portion are both plausible since they both preserve the notion of at limit at $c$ [keep in mind that the hyperbola would preserve the limit if of odd degree]. This should help you answer your question.