Properties of Cumulative Distribution Function

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X$ be the cumulative distribution function.

That is:


 * $\map {F_X} x = \map \Pr {X \le x}$

for each $x \in \R$.

Then $F_X$ has the following properties: