The statement is stated by our teacher as follows :
Suppose $$\lim_{x\rightarrow x_{0}}f'(x)=a\in{\mathbb R} $$ then $$f'(x_{0})=a$$
, here we do not assume $f'(x)$ is continuous at $x_{0}~.$ The statement is so strange to me . But our teacher claims that this exercise can be found in the text book of Apostol
For $h\ne 0$, then by Mean Value Theorem we have $\dfrac{1}{h}(f(x_{0}+h)-f(x_{0}))=f'(\eta_{h})$, where $\eta_{h}$ is strictly in between $x_{0}$ and $x_{0}+h$, in particular, $\eta_{h}\ne x_{0}$. Taking $h\rightarrow 0$, then $\eta_{h}\rightarrow x_{0}$, use the assumption that $\lim_{x\rightarrow x_{0}}f'(x)=a$, where we note that $\eta_{h}\ne x_{0}$, so $\lim_{h\rightarrow 0}f'(\eta_{h})=a$, hence $f'(x_{0})=a$.
Given $\epsilon>0$, there is a $\delta>0$ such that $0<|x-x_{0}|<\delta$, $|f'(x)-a|<\epsilon$, then whenever $0<|h|<\delta$, we have $0<|\eta_{h}-x_{0}|<\delta$ and hence $|f'(\eta_{h})-a|<\epsilon$.
