## Corollary 6.3.7: Discontinuities of Second Kind

If

*f*has a discontinuity of the second kind at*x = c*, then*f*must change from increasing to decreasing in every neighborhood of*c*.### Proof:

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.