I have some problems with the following exercise, maybe due to alack of knowledge:
Let $M$ be a connected smooth manifold and let $$ f \colon M \to N$$ be an analytic map. Denote by $C_f \subset M$ the set of critical points of $f$. How can I prove that if $C_f\neq M$, then $f^{-1}(f(C_f))$ is a null set?
I tried to find relations between some topological aspect of a null set but without any results. By Sard's Theorem $f(C_f)$ is a null set, and it seems reasonable that I have to use the fact that I have to extend my analytic function in order to solve the problem. But in general I don't know how to proceed. Keep in mind that my knowledge in measure theory is very limited :(
Using the analyticity of the $f$ I was able to prove that $f(C_f)$ is at most countable, but I don't know how to translate it to
Thanks in advance!