Let $g : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function whose derivative is continuous, and such that $g ( g ( x ) ) = x$ for all $x > 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.
Let $g : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function whose derivative is continuous, and such that $g ( g ( x ) ) = x$ for all $x > 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.