Classification of discontinuities

While continuous functions are important in mathematics, not all functions are continuous. If a function is not continuous at a limit point (also called an "accumulation point" or "cluster point") of its domain, 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.

In elementary real analysis, discontinuities of real functions of one real variable are often distinguished according to the behavior of one-sided limits. While a classification is not entirely standard, a common division is between discontinuities of the first kind, where the relevant one-sided limits exist, and discontinuities of the second kind, where at least one one-sided limit fails to exist or is infinite. Special cases include removable discontinuities and jump discontinuities.