Conjecture: Let $f:\mathbb{R}→\mathbb{R}$ be an everywhere differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ almost everywhere and $f'(0)=0$. Then is f necessarily a constant function?

Can you give me a counter-example ?

I have already asked the question here https://math.stackexchange.com/questions/2859868/differential-equation-changing-sign-almost-everywhere