Recall that a point x=c is a discontinuity of the first kind if and exist. Suppose that f(x) is monotonically increasing (an analogous argument will exist for monotonically decreasing f(x)). Then for an increasing infinite sequence of with , we see that is an increasing sequence. Consider this in combination with the boundedness of f(x), and what that says about the existence of . Similar work for a decreasing sequence , will give a similar result for .

--Kevin C.