discontinuity

Discontinuity refers to a point at which a function is not continuous. A function f is continuous at a point a if the limit of f(x) as x approaches a equals f(a). If that condition fails, a is a point of discontinuity. Discontinuities can occur at isolated points or along entire intervals, depending on the function.

In real-valued functions on the real line, several common types are recognized. A removable discontinuity occurs when the limit L = lim x→a f(x) exists but either f(a) is different from L or f is undefined at a. Redefining f(a) to be L makes the function continuous at a. A jump discontinuity happens when the left-hand limit and the right-hand limit exist but are not equal. An infinite or unbounded discontinuity occurs when one-sided limits diverge to infinity or negative infinity. An oscillatory discontinuity arises when the limit does not exist due to persistent oscillation, as in f(x) = sin(1/x) near x = 0.

Discontinuities are central in analysis because they mark where intuitive notions of smooth behavior fail. They can be identified by examining limits from either side, the value of the function at the point, or the behavior of the function in a neighborhood. In higher mathematics, discontinuities can also be described topologically as points where the function fails to be continuous in the given domain, or, in complex analysis, as poles or essential singularities.

Applications appear across mathematics, physics, and engineering. Discontinuities model abrupt changes such as shocks in fluid dynamics, phase boundaries in materials, or piecewise definitions in computational algorithms.