Corollary 6.3.7: Discontinuities of Second Kind
Suppose not, i.e. f has a discontinuity of the second kind at a point x = c, and there does exist some (small) neighborhood of c where f, say, is always decreasing. But then f is a monotone function, and hence, by the previous theorem, can only have discontinuities of the first kind. Since that contradicts our assumption, we have proved the corollary.