A function with a condition This question was asked for one of our internal tests (high school)

$\mathit f(x):$ $ $$\Bbb R  \mapsto \Bbb R $ is a continuous function and has the property $\;\mathit f(f(x)) = 1-x \,\,$$\,\forall \,\, x \in [0,1] $ and $\mathit J= \int _0 ^1 f(x) dx$, then find the value of $\mathit 1/J.$

Will someone provide an example of such a function.
NB: I do not require a solution, please provide me only an example of such a function.

Hints:

*

*Show that $f(x)$ must be bijective and, therefore, strictly increasing or decreasing.

*Show that $f(f(x))$ is strictly increasing no matter if $f(x)$ is increasing or decreasing.

*Observe a contradiction since $1-x$ is strictly decreasing. So there is no such function.

