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$f$ is a differentiable function such that $f(f(x)) = x$ for $x ∈ [0, 1]$. And $f(0) = 1$. What is the value of $\int_0^1{(x − f(x))^{2016}}dx$ ?

I have noticed that one solution of $f$ is $f(x) = 1-x$, then the integral can be easily evaluated. Can it be done without assuming $f(x) = 1-x$?

Archisman Panigrahi
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  • This looks like a contest problem. If so, there is a tag for that, and it could be a good gesture to give the reference. – mickep May 08 '17 at 11:58

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