Elementary Classical Analysis, Marsden, 2nd ed, Chapter 6, Problem 15
Problem
If $f:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable. Assume there is no $x \in \mathbb{R}$ such that $f(x)=0=f'(x)$. Show that $S=\{x\;|\;0\leq x \leq 1,f(x)=0\}$ is finite.
Answer
Note that $S=f^{-1}(0)$ is closed since $f$ is continuous and $\{0\}$ is closed. Also, $S$ is bounded since $S \subset [0,1]$. These imply $S$ is compact. If $S$ is infinite, there is a sequence $\{x_n\} \subset S$ and its subsequence $x_{n_k} \rightarrow x \in S$. Then
$$ f'(x) = \lim_{k \rightarrow \infty} \frac{f(x) - f(x_{n_k})}{x-x_{n_k}} =0. $$
Hence, $S$ has to be finite.
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