Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function?
For instance:
$X$ 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
Best Answer
A simple example is $X$ uniformly distributed on the usual Cantor set, in other words, $$X=\sum_{n\geqslant1}\frac{Y_n}{3^n},$$ for some i.i.d. sequence $(Y_n)$ with uniform distribution on $\{0,2\}$.
Other examples are based on binary expansions, say, $$X=\sum_{n\geqslant1}\frac{Z_n}{2^n},$$ for some i.i.d. sequence $(Z_n)$ with any nondegenerate distribution on $\{0,1\}$ except the uniform one.
These distributions have no density with respect to Lebesgue measure, even partly, since $P(X\in C)=1$ for some Borel set $C$ with zero Lebesgue measure. They have no atom either, in the sense that $P(X=x)=0$ for every $x$.