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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.

Answers