The probability cumulative distribution must be monotone and 

A cumulative distribution function (CDF) is defined as:

\(P\;\left( {Z < z} \right) = \mathop \sum \limits_{ - \infty }^z f\left( z \right) = F\left( z \right)\)

which is the probability that Z is less than or equal to some specific z, i.e. it defines a cumulative sum of up to a specified value of z.

Also, f(z) is the probability density function.

Since the probability is always ≥ 0.

So, CDF is the summation of the Probabilities of discrete random variables. So CDF is always non-decreasing with finitely many jump-discontinuities. 

Mathematically it is defined as F_Z[zi] = P[Z ≤ zi] 

Consider the example of dice. Let, X is the random variable. 

The distribution function is shown below