(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Assume that [tex] f:[a,b] \rightarrow \mathbb{R} [/tex] is continuously differentiable. A critical point of f is an x such that [tex] f'(x) = 0 [/tex]. A critical value is a number y such that for at least one critical point x, y = f(x). Prove that the set of critical values is a zero set.

2. Relevant equations

A set Z is said to be a zero set if for each [itex] \epsilon [/itex] there is a countable covering of Z by open intervals [tex] (a_i , b_i) [/tex] such that [itex] \sum_{n=1}^\infty} b_i - a_i \leq \epsilon [/itex]

3. The attempt at a solution

I'm not even sure really where to start with this one. Normally when I start a problem, I always check to make sure that it intuitively makes sense. If we let C be the set of critical points, then the goal is to show f(C) has measure zero, but I really don't see how to approach this problem. I don't really see how to use the fact that the derivative is continuous. I'd love a hint in the right direction to start or something, as this has me pretty baffled. Thanks!

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# Homework Help: Smooth function's set of critical values

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