Why “supremum”, not “maximum” in Kolmogorov-Smirnov test?

Question

Why the test function in Kolmogorov-Smirnov test takes the supremum of the set of differences, not maximum? When maximum (greatest element) of a set does not exist then supremum may exist, but if greatest element exist then supremum is same as greatest element. I want to know why here greatest element does not exist?

Thanks
Prasenjit

Answer 1

Suppose a difference is an increasing continuous function on open interval and zero everywhere else. Then the maximum will not exist, but supremum will.