discontinuous

Discontinuous is a term used in mathematics and related fields to describe a property where a function, sequence, or relation fails to be continuous at some point or over a domain. It is the negation of continuity and can refer to a single point or to entire regions.

In real analysis, a function f is discontinuous at a point c if the limit of f(x) as x approaches c either does not exist or exists but is not equal to f(c). Discontinuities are commonly classified into removable, jump, and infinite types. A removable discontinuity occurs when a function can be redefined at a point to make it continuous. A jump discontinuity happens when the left and right limits exist but are different. An infinite discontinuity involves limits that diverge to infinity or negative infinity.

Examples help illustrate these ideas. The Heaviside step function has a jump discontinuity at 0. The Dirichlet function, which is 1 at rationals and 0 at irrationals, is discontinuous everywhere. The function f(x) = sin x / x has a removable discontinuity at x = 0 if not appropriately defined there, while f(x) = 1/x has an infinite discontinuity at x = 0. The floor function has jump discontinuities at every integer.

In topology and analysis, continuity is defined in terms of preserving limits or the preimage of open sets. A function is discontinuous if there exists a point or region where this preservation fails. Discontinuities influence numerical methods, integration, and differentiation, and they often model abrupt changes in applied contexts such as physics, engineering, and signal processing.