Based on the concept that $f$ is increasing then all the solutions of the equation $f(x)=f^{-1}(x)$ lie on the line $y=x$. I have two questions:
Can I solve $f(f(x))=x$ by taking $f(x)=x$ if $f$ is monotonic increasing?Answer to this seems yes to me.
What are the necessary and sufficient condition for $f(f(x))=x$?Is is necessary for it have an inverse or monotonic?
Can we not have an invertible function which satisfies $f(f(x))=x$