How can a function fail to be continuous? There are several ways for f(x) to be discontinuous at x=a: point/removable discontinuity, jump/step discontinuity, and essential/infinite discontinuity (potential problems listed from "least serious" to "most serious"). Note: there is some ambiguity in the definition of discontinuity. Sometimes discontinuities are meant to include singularities (where f(a) is undefined); sometimes the concept of discontinuities is limited to points where f(a) is defined.