piecewisecontinuous

Piecewise continuous describes a class of functions that are continuous on each member of a finite collection of subintervals of their domain, while allowing finitely many jump discontinuities at the subinterval endpoints. Formally, an interval I with a function f: I -> R is piecewise continuous if there exists a finite partition a = x0 < x1 < ... < xn = b of I such that f is continuous on each open subinterval (xi-1, xi) for i = 1,...,n, and the one-sided limits lim_{x→xi-} f(x) and lim_{x→xi+} f(x) exist and are finite for i = 1,...,n-1. In many treatments the values of f at the partition points are chosen to be compatible with the surrounding limits, so that f has well-defined left and right limits at each xi.

If a function is defined on an interval and has only a finite number of discontinuities, it is often described as piecewise continuous on that interval. Piecewise continuous functions are locally Riemann integrable, and on a finite closed interval [a,b], a piecewise continuous function is Riemann integrable. The concept is widely used in analysis and applied mathematics because such functions can be integrated by splitting the integral at the points of discontinuity, and they commonly arise as solutions to problems with piecewise-defined rules.

Examples include f(x) = x for x ≤ 0 and f(x) = 2x for x > 0, which has a jump at 0; or f(x) = sin x, which is continuous everywhere and thus piecewise continuous. Functions with infinitely many discontinuities, such as the floor function on the real line, are not piecewise continuous under the common finite-partition definition.

See also: Discontinuity, Riemann integrability, piecewise function, piecewise smooth.