This appeared as a throwaway statement in a proof - that a strictly monotonic (increasing) transformation of a continuously distributed random variable (I am assuming that this means that the distribution function is continuous, not that the random variable is absolutely continuous) is also a continuously distributed random variable.
So the setup $X:\left(\Omega, \mathscr{F}, \mathbb{P}\right) \longmapsto \left(\mathbb{R}, \mathscr{B}(\mathbb{R}), \mathbb{P}_X\right)$ and $h: \mathbb{R} \longmapsto \mathbb{R}$ and $h(X) = Y$, where clearly $h$ needs to be measurable. So the claim is that if $h$ is monotonic, then $Y$ is a continuously distributed random variable.
A proof or a reference to a textbook would be appreciated.