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Putnam 2010 A2

Find all differentiable functions $f: \Bbb{R} \to \Bbb{R}$ such that $\displaystyle f^\prime(x)=\frac{f(x+n)-f(x)}{n}$ for all $x \in \Bbb{R},\ \forall n \in \Bbb{Z}_+^*$.

Putnam 2010 A2

Hint: Prove that $f^\prime$ is periodic of period $1$.