holomorphic function with real arguments smooth?

Question

suppose we have a holomorphic function $f:\mathbb{C}\rightarrow \mathbb{C}$ and we have $f(x)\in\mathbb{R}$ for all $x\in\mathbb{R}$. Is it true that $f_{\vert\mathbb{R}}:\mathbb{R}\rightarrow \mathbb{R}$ must be a smooth function?

I think the answer is yes, but I am not sure. Does someone know how one would prove it?

Best regards

Answer 1

Proof sketch:

For functions of a complex variable, holomorphic = analytic. That means that this function has a convergent power series everywhere it is holomorphic, including all points of $\mathbb R$. This shows $f_{|\mathbb R}$ is analytic at all points of $\mathbb R$, and thus is infinitely differentiable everywhere.