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A characterization of the zero function

Source: Romanian National Olympiad, grade xi, p.1

August 23, 2019
functionreal analysis

Problem Statement

Find all differentiable functions f:RR f:\mathbb{R}\longrightarrow\mathbb{R} that verify the conditions: \text{(i)} \forall x\in\mathbb{Z}   f'(x) =0
\text{(ii)} \forall x\in\mathbb{R}  f'(x)=0\implies f(x)=0