I found the following the example of a cumulative distribution function:
Suppose that $X$ takes only the discrete values $0$ and $1$ with an equal probability on the interval $[0,1]$. Then the CDF of $X$ is given by
$F(x) = \begin{cases} 0 &:\ x < 0\\ 1/2 &:\ 0 \le x < 1\\ 1 &:\ x \ge 1. \end{cases} $
However, I do not really understand how one reaches this result. The cases for $x \lt 0$ and $x \ge1$ follow by definition. However the case $0 \le x \lt 1$ appears to be rather obscure:
E.g. $X$ is uniformly distributed on $[0,1]$ and comparing these two distributions, then $F(x)$ is bigger given the non-discrete distribution than in the uniform distribution for $0 \le x \lt 0.5$.
I don't have a proper understanding yet, how to arrive at above's CDF. I would be very happy, if someone could enlighten me.