Find all continuous function such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$ for all $x, y\in\mathbb{R}$. Denote $\{x\}=x-[x]$ in which $[x]$ is the largest integer number does not exceed x.
Find $f$ such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$
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functional-equations