Is f necessarily constant?

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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