Assume $\displaystyle f:R \rightarrow R$ has a finite derivative everywhere on $\displaystyle (a,b)$ except possibly at a point $\displaystyle c \in (a,b)$. If we have $\displaystyle lim_{x \rightarrow c} f'(x)=A$, then prove $\displaystyle f'(c)$ exists and is equal to $\displaystyle A$.