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Is f necessarily constant?

January 29, 2017 Tolaso

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f'(x)=0$ forall $x \in \mathbb{Q}$ . Does it follow that $f$ is necessarily constant?

Solution

Since $f'(x)=0$ forall $x \in \mathbb{Q}$ it follows that $f(x)=c$ forall $x \in \mathbb{Q}$ . Let $x_0 \in \mathbb{R} \setminus \mathbb{Q}$ and suppose that $f(x_0) \neq c$ . Since $\mathbb{Q}$ is dense there will exist a sequence $\{q_n\}_{n \in \mathbb{N}}$ of rational numbers that it converges to $x_0$ . Thus:

$\displaystyle c \neq f\left ( x_0 \right ) = \lim_{x\rightarrow x_0} f(x)= \lim_{n \rightarrow +\infty} f \left ( q_n \right ) = c$

contradicting what we had assumed in the first place. Hence $f$ is constant.

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