Analysis in $\mathbb{Q}$

Calculus Level 3

It is a well-known theorem that if $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function such that $f'(t)=0$ for all $t\in\mathbb{R}$, then $f$ is constant.

Is it true that if $f:\mathbb{Q}\to\mathbb{Q}$ is a differentiable function such that $f'(t)=0$ for all $t\in\mathbb{Q}$, then $f$ is constant?

Notations:

• $\mathbb R$ denotes the set of real numbers.
• $\mathbb Q$ denotes the set of rational numbers.
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