Questions regarding concept of Derivative

Question

There are 2 question I come across
1)Assume f has finite derivative at each point of open interval (a,b).Assume that $\lim_{x\to c} f'(x)$ exists and is finite for some interior point c .Then prove that value of limit must be $f'(c)$
Difficulty regarding this question is that I think Derivative of function at point is limit .Here with statement state as f has finite derivative so derivative limit must exist at all interior point .So it is trivial that f'(c)=$\lim_{x\to c}f'(x)$ .
Or Is this question wanted to prove that at c derivative is continuous.I had difficulty regarding this concept . 2) Let f is continuous on (a,b)with a finite derivative f' everywhere in (a,b)except at c .If $\lim_{x\to c}f'(x)$ exist and has value A then show that f'(c) must also exist and value A.

I had same problem in both of this question as what is difference between f'(c) and $\lim_{x\to c}f'(x)$ Any help will be appreciated .Please help me out I was not able to capture this basis notion I had read many book Still this problem Persist.
Thanks

Answer 1

Let $d=\lim_{x\to c}f'(x)$; you want to prove that $d=f'(c)$, that is, that$$\lim_{h\to0}\frac{f(c+h)-f(c)}h=d.$$Take $\varepsilon>0$. There is a $\delta>0$ such that $|h|<\delta\implies\bigl|f'(c+h)-d\bigr|<\varepsilon$. So, if $|h|<\delta$, then, by the mean value theorem, $\frac{f(c+h)-f(c)}h=f'(c^\star)$ for some $c^\star$ between $c$ and $c+h$; in particular $c^\star=c+h^\star$ for some $h^\star$ such that $|h^\star|<|h|<\delta$ and so$$\left|\frac{f(c+h)-f(c)}h-d\right|=\bigl|f'(c+h^\star)-d\bigr|<\varepsilon.$$