Continuity

Continuity is a fundamental idea in mathematics describing the property that small changes in input lead to small changes in output. In analysis, a function f is continuous at a point a if for every ε>0 there exists a δ>0 such that |x−a|<δ implies |f(x)−f(a)|<ε. If this holds for every point a in a domain, f is continuous on that domain. Equivalently, the limit of f(x) as x approaches a equals f(a).

In topology, continuity is defined more generally: a function between topological spaces is continuous if the preimage of every open set is open. In metric spaces this reduces to the epsilon–delta condition above and is equivalent to preserving limits of convergent sequences (sequential continuity).

Related notions include uniform continuity, where δ can be chosen independently of the point a, and continuity on a compact domain, which implies uniform continuity. Discontinuities are points where continuity fails; common types include removable, jump, and essential discontinuities.

Continuity also appears in physics and engineering, where it describes smooth behavior of physical fields and the use of continuity equations, such as conservation of mass in fluids.

Historically, continuity was developed in the 19th century by Bolzano, Cauchy, and Weierstrass, among others, to formalize limiting processes that underlie calculus and topology.