Is this enough to be differentiable?

Solution 1:

the 'one sided' part of this is a standard and often asked for result. It follows from the mean value theorem (e.g. for $x>c$): $$\frac{f(x)-f(c)}{x-c} = f^\prime(\xi) $$ for some $\xi\in (c, x)$ - now let $x\rightarrow c$ which then implies that the r.h.s of this equation converges (by assumption), which in turm implies one-sided differentiability in $c$. If the limits for $x\rightarrow c$ of $f^\prime(x)$ don't coincide from the left and right then $f $ will not be differentiable, otherwise yes...

(Note that the mean value theorem only requires differentiability in the open interval $(c, x)$).