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Possible values for a sum under some functional equation

Source: Bundeswettbewerb Mathematik 2018, Round 2, Problem 2

February 5, 2019
functionalgebraalgebra proposedfunctional equationSum

Problem Statement

Consider all functions f:R→Rf:\mathbb{R} \to \mathbb{R} satisfying f(1−f(x))=xf(1-f(x))=x for all x∈Rx \in \mathbb{R}. a) By giving a concrete example, show that such a function exists. b) For each such function define the sum Sf=f(−2017)+f(−2016)+⋯+f(−1)+f(0)+f(1)+⋯+f(2017)+f(2018).S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018). Determine all possible values of SfS_f.
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