One point differentiable function.

Question

My question is pretty clear : does there exist some function which is differentiable only at one point and defined on all $\mathbb{R}$?

I proved that there are no such functions, but suddenly thought about the function which return $x$ on $\mathbb{R} / \mathbb{Q}$ and returns zero on all other numbers.

What do you think about it?

Answer 1

That function is continuous at $0$ but not differentiable. Replace $x$ with $x^2$, and the resulting function is differentiable at $0$ as well. Use the limit definition of derivative at a point and the Squeeze Theorem.