A function not differentiable at a point but whose derivative has a limit

Question

Does there exist a function $f:\mathbb{R}\to\mathbb{R}$ such that

1. $f$ is continuous on a neighborhood of $0$,
2. $f$ is differentiable at all $x$ close to $0$ except at $0$ itself, and
3. $\lim_{x\to 0} f'(x)$ exists, but
4. $f$ is not differentiable at $0$?

Answer 1

By the mean value theorem $$ \frac{f(x)-f(0)}{x} = f'(\xi_x) $$ for some $\xi_x$ between $0$ and $x$. By assumption $f'(\xi_x) \to L$ for some $L$ as $x \to 0$. Thus $f'(0)$ exists and equals $L$.

Answer 2

Are you allowing $\lim_{x\to 0} f'(x)$ to be infinite? In that case, $f(x) = x^{1/3}$ is an example. But if by ''$\lim_{x\to 0} f'(x)$ exists'' you imply that the limit is finite, then see the answer given by nullUser.