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*}