Problem on Definite Integration and functions

Question

Q. The function f is continuous and has the property $f(f(x))=1-x$ for all $x\in [0,1]$ and $$J = \int_{0}^{1} f(x) dx$$ then find $J$.

My Attempt- I have no clue to this problem! Instead I tried reorganising the functional equation as $$f(x) = 1- f^{-1}(x) $$,but I don't know any way to proceed from here. So am I doing good till now ? Or I have to do something else? Do you have any hint?

Any help would be appreciated

Answer 1

You could for example have $f(x) = x+m$ for $0 \le x \le 1$ and $f(x) = m + 1 - x$ for $m \le x \le m+1$, with $m > 1$ or $m < -1$, and $J = m + 1/2$. So without some additional assumptions, $J$ is not determined.

Answer 2

There is no such function: let $f(x)=f(y)$. Apply $f$ on both sides to get $1-x=1-y$, so $x=y$. Hence $f$ is one-to-one. Any one-to-one continuous function on strictly monotonic This makes $f\circ f$ strictly increasing but $1-x$ is strictly decreasing. Hence there is no such function $f$.