DE-BUNDESWETTBEWERB 2018 Problem 6

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