Definition of a distribution function

A function $F: \mathbb{R} \mapsto [0,1]$ satisfying the following properties is a distribution function.

1. $F$ is right continous;
2. $F$ is monotone non-decreasing,
3. $F$ has limits at $\pm\infty$
   $$\begin{align*} F(\infty) &:= \lim_{x\uparrow \infty} F(x) = 1 \\ F(-\infty) &:=\lim_{x\downarrow - \infty} F(x) = 0 \end{align*}$$