I am interested in analyzing the properties of the following function: $$a(z)=Pr\left(g(X) \leq z \right),$$ where r.v. $X$ is absolutely continuous with respect to the Lebesgue measure in $\mathbb{R}$, and $g(\cdot): \mathbb{R} \rightarrow \mathbb{R} $.
It is trivial of course to show the following results: (i) if $g$ is strictly monotone and continuous, then function $a(\cdot)$ is continuous; (ii) if $g$ is strictly monotone and the derivative is not zero a.e., then function $a(\cdot)$ is differentiable a.e..
My question is: are there weaker conditions on $g$ that will guarantee the continuity and differentiability properties of $a(\cdot)$?