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Functional equation

Source: Iranian TST 2018, second exam day 1, problem 1

April 15, 2018
functionalgebrafunctional equation

Problem Statement

Find all functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} that satisfy the following conditions: a. x+f(y+f(x))=y+f(x+f(y))   \forall x,y \in \mathbb{R} b. The set I={f(x)−f(y)x−y∣x,y∈R,x≠y}I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\} is an interval.
Proposed by Navid Safaei
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