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 $0$?