Can any smooth even function $f: \mathbb{R} \to \mathbb{R}$ be written as a smooth function of $x^2$?

Question

If $f: \mathbb{R} \to \mathbb{R}$ is a smooth ($C^\infty$) even function, then the Taylor expansion at any point contains only even powers of $x$. Is it true that $f$ can be written as a smooth function of $x^2$? Similarly, if $f$ is an odd function, the Taylor expansion around any point has a factor of $x$ that can be factored out. In this case, can $f$ always be written as $x\cdot g(x)$ for some smooth $g$?

Answer 1

Yes.

This is a theorem by Whitney, have a look to this page.