Monotone mean-value point of an integral average ($f(g(x))=x^{-1}{\int }_0^{x}f(t)dt$)

For a continuous strictly decreasing real function, its average $h(x)=x^{-1}{\int }_0^{x}f(t)dt$ lies strictly between $f(x)$ and $f(0)$ for $x>0$. There is a unique $g(x)\in (0,x)$ with $f(g(x))=h(x)$. If $f$ is differentiable and $f^{\prime }<0$ throughout the positive half-line, the fundamental theorem of calculus and the derivative formula for the monotone inverse giveContinuity of $f^{\prime }$ is unnecessary: differentiability of a strictly monotone function with nonzero derivative at a point suffices for differentiability of its inverse there.