TITLE: If $f^2$ and $f^3$ are smooth, does it follow that $f$ is smooth?
QUESTION [5 upvotes]: Let $f: \mathbb{R} \to \mathbb{R}$ be given. Assume that the square and cube of $f$ are smooth. Is $f$ smooth? That is if $f \cdot f \in C^{\infty}$ and $f \cdot f \cdot f \in C^{\infty}$, does it follow that: $f \in C^{\infty}$

I got this from another question on SE. 
So, $f(x)^2$ is infinitely differentiable, and so is: $f(x)^3$ 
Also, realize:
$$f(x) = \frac{f(x)^3}{f(x)^2}$$
But what can I do?

REPLY [2 votes]: The answer is Yes, and the proof is quite technical:
See the answer to this question: https://mathoverflow.net/questions/125861/f3-f2-are-the-cube-and-quadratic-of-f-respectively-and-both-infinite-differe