Continuity of CDF of a discrete Random Variable

Question

Let $X$ be a discrete random variable with $Range(X) \subset \mathbb{N}$. How to prove that the CDF is

a) not continuous at all points in the range of $X$.

b) differentiable at all other points.

Answer 1

HINT

Let $p_n = \mathbb{P}[X=n]$ and consider some $n$ with $p_n \ne 0$. The CDF is defined as $$ F_X(x) = \mathbb{P}[X \le x] $$ so as you are approaching from the left would be $$ F_X(n^-) = \mathbb{P}[X < n] = \mathbb{P}[X \le n-1] $$ and when you approach from the right you get $$ F_X(n^+) = \mathbb{P}[X \le n] = F_X(n^-) + p_n. $$ Can you complete the problem now?