Let $X$ be a random variable with support $(a,b)$ and probability distribution $F$ twice continuously differentiable (density $f$). Define $g(y):=\mathrm{E}[X\,|\, X>y]$.
I'm looking for a sufficient condition under which $g^{-1}$ exists. I think that $f(y)>0$ for all $ y \in (a,b)$ is sufficient, but could not prove it. Can you find a sufficient condition?
Knowing $f(y) > 0$ is indeed a sufficient condition: It implies that $g$ is strictly monotonically increasing, which is a sufficient condition for $g$ to be invertible.
You can show that $g$ is strictly monotonically increasing by showing that $g'(y) >0$ for all $y$.