I'm having some struggles with an aspect about something apparently trivial about Sard's theorem, but couldn't find anything online.
Let $f$ be a polynomial.
According to Sard's theorem, the image $f(Z)$ of the set of critical values $$Z = \{a \in X : f'(a) = 0\}$$ has measure zero.
What if I want to show that the set $Z$ itself has measure zero in the domain of $f$?
I feel like it's so simple but i just can't get behind it.
In this case Sard's theorem is not needed. If $p$ is a nonzero polynomial, its set of zeros is finite by induction on its degree. If $f$ is not constant, then the set $Z$ of critical points is therefore also finite (it is the set of zeros of the polynomial $f'$), so it has measure zero!
Edit: @Captain Lama has suggested that you may be dealing with a polynomial in more than one variable. In this case, we cannot deduce that the set of zeros of $f'$ is finite, but we can still show that it has measure zero. You can do this with Fubini's theorem and induction on the number of variables – see here for an algebraic proof. You do need to make sure that $f$ is not constant, of course!