Real Analysis

Let $f\colon \mathbb{R}\to \mathbb{R}$ be an everywhere-differentiable function such that $f'=f$ on rationals, i.e. $f'(q) = f(q)$ for all $q\in \mathbb{Q}$. Must $f(x)=ce^x$ identically for some constant $c$?

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[Source: Communicated by Mark Sellke]