Is there r.v. that are neither discrete nor continuous ? My definition of continuous r.v. is that $x\longmapsto f_X(x)$ (the mass function) is continuous and the r.v. is discrete if there is a countable set $\mathcal D$ s.t. $\mathbb P\{X\in \mathcal D\}=1$. So is there r.v. s.t. $f_X$ is not continuous but there is no countable set $\mathcal D$ s.t. $\mathbb P\{X\in \mathcal D\}=1$.
For example something as $\mathbb P\{X\in \mathcal C\}=1$ where $\mathcal C$ is the Cantor set, but there is no countable subset $\mathcal D$ of $\mathcal C$ s.t. $\mathcal P\{X\in \mathcal D\}=1$, but $f_X$ is not continuous. Does such r.v. exist ? If yes, which one ? If no, why ?
Or just think of a random variable that is $=2$ with probability $\frac12$ and is taken from a unifmormly distributed varable in $[0,1]$ otherwise ...