RO-NMO 2004 Problem 121

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have $$n^2 \int_{x}^{x + \frac{1}{n}} f(t) dt = n f(x) + \frac12 .$$

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