Let $f:(0,\infty)\rightarrow\mathbb{R}$ be a continuous function and let $F$ be its antiderivative. Suppose there exists a finite limit $\lim_{x\to 0} F(x)$. We can extend $F$ to a continuous function by defining $F(0):=\lim_{x\to 0} F(x)$. The question is: Is $F$ differentiable at $0$?
When $f$ has a limit $\alpha=\lim_{x\to 0}f(x)$ by l'Hospital' rule we have
$$\lim_{x\to 0}\frac{F(x)-F(0)}{x}=\lim_{x\to 0}F'(x)=\lim_{x\to 0}f(x)=\alpha$$
which means $F$ is differentiable when this limit is finite.
The problem is when $f$ does not have a limit as $x$ tends to $0$. In this case there are examples of functions when $F$ is differentiable (take $F$ which is differentiable but the derivative is not continuous at $0$), but I haven't found an example when it is not differentiable. That's why I'm looking for a proof or a counterexample.
Not necessarily.
Take $f(x)= (-\frac{\cos(1/x)}{x})+\sin(1/x).$ Its antiderivative is $x \sin(\frac{1}{x})$, which can be continuously extended to $0$, but is not differentiable there.