Science:MATH105 Probability/Lesson 2 CRV/2.02 CDF Properties
CDFs share many of the same properties for both the continuous and discrete cases. In the following theorem, the three properties listed are common to both the discrete and continuous cases.
Theorem: Properties of a Distribution Function 

If F(x) is a cumulative distribution function for the random variable X, then

It may help the reader at this point to recall the definition of a "nondecreasing function": a function, f(x), is a nondecreasing function if f(x_{1}) ≤ f(x_{2}) for all x_{1} and x_{2} in the domain of f, and x_{1} < x_{2}.
Proofs for the first two property are beyond the scope of MATH 105. For the interested reader, a proof of the third property is below.
Proof of The Third Property
In general, for any interval of finite length Δx, there is a nonnegative probability that X lies somewhere in that interval, no matter where our interval lies or how small we make Δx. Therefore, if we were to choose any interval [x , x + Δx], the probability that our continuous random variable X lies inside of this interval,
must be nonnegative:
But this can be expressed as the difference
Rearranging yields, for all ∆x,
This is the definition of a nondecreasing function. Therefore, the CDF, which is defined as F(x) = Pr(X ≤ x), is a nondecreasing function of x.