Discontinuities

Discontinuities are points at which a function is not continuous. In real analysis, a function f is continuous at a point x0 if the limit of f(x) as x approaches x0 equals f(x0). The set of points where this condition fails is the discontinuity set. Discontinuities can occur in many contexts, including real-valued functions of a real variable, complex functions, and functions between more general topological spaces.

In real-valued functions of a real variable, discontinuities are commonly classified as removable, jump, infinite, or oscillatory. A removable discontinuity occurs when the limit as x approaches x0 exists but either f(x0) is not defined or f(x0) differs from the limit. For example, f(x) = (x^2 − 1)/(x − 1) for x ≠ 1 has limit 2 as x → 1, so redefining f(1) = 2 removes the discontinuity. A jump discontinuity occurs when the left and right limits exist but are unequal, as in f(x) = floor(x) at integer points. An infinite discontinuity occurs when the limit diverges to infinity, as in f(x) = 1/x at x = 0. An oscillatory discontinuity happens when the limit does not exist due to persistent oscillation, such as f(x) = sin(1/x) as x → 0.

Some structural results accompany discontinuities. For monotone real-valued functions, all discontinuities are jump discontinuities and form a countable set. In higher dimensions, discontinuities can occur along curves or more complex sets, and the concept of continuity is tied to the domain’s topology. Discontinuities are central in analysis, numerical methods, signal processing, and physics, where they often model abrupt changes.