Continuous functions are of utmost importance in
mathematics, functions and applications. However, not all
functions are continuous. If a function is not continuous at a point in its
domain, one says that it has a
discontinuity there. The set of all points of discontinuity of a function may be a
discrete set, a
dense set, or even the entire domain of the function. This article describes the
classification of discontinuities in the simplest case of functions of a single
real variable taking real values.