RO-NMO 2005 Problem 92

Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which $$x(f(x+1)-f(x)) = f(x),$$ for all $x\in\mathbb{R}$ and $$| f(x) - f(y) | \leq |x-y| ,$$ for all $x,y\in\mathbb{R}$ .

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