CDF is:
# Monotonically non-decreasing for each of its variables,
# Right-continuous in each of its variables,
#
# and for all .

Not every function satisfying the above four properties is a multivariate CDF, unlike in the single dimension case. For example, let for or or and let otherwise. It is easy to see that the above conditions are met, and yet is not a CDF since if it was, then as explained below.

The probability that a point belongs to a hyperrectangle is

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