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Find all continuous functions

Source: 2019 Jozsef Wildt International Math Competition-W. 27

May 18, 2020

integrationfunctional equationfunction

Problem Statement

Find all continuous functions

f : \mathbb{R} \to \mathbb{R}

such that

f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}