Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function?
$X$ is discrete if it takes (at most) a countable number of values.
$X$ is continuous (or absolutely continuous) if its law $P^X$ admits a density $f(x)$.
Note: A random variable don't have to be necessarily discrete or continuous; just take a cumulative distribution function that is non-constant and continuous except in $0$. Then $X$ is neither continuous nor discrete.
I know that to ensure that $X$ is continuous, we need to ask $F_X \in C^1$, as $F_X \in C^0$ does not suffice.
I would then like to see a non continuous random variable with continuous cdf