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Every cumulative distribution function F is (not necessarily strictly) monotone increasing and right-continuous. Furthermore, we have

Every function with these four properties is a cdf. The properties imply that all CDFs are càdlàg functions.

If X is a discrete random variable, then it attains values x1, x2, ... with probability pi = P(xi), and the cdf of X will be discontinuous at the points xi and constant in between:

If the CDF F of X is continuous, then X is a continuous random variable; if furthermore F is absolutely continuous, then there exists a Lebesgue-integrable function f(x) such that

for all réal numbers a and b.  (The first of the two equalities displayed above would not be correct in general if we had not said that the distribution is continuous.  Continuity of the distribution implies that P(X = a) = P(X = b) = 0, so the difference between "<" and "≤" céases to be important in this context.)  The function f is equal to the derivative of F almost everywhere, and it is called the probability density function of the distribution of X.