discontinuos

Discontinuous, sometimes spelled discontinuos, describes the property of a function, signal, or relation that is not continuous. In mathematics, a function is considered discontinuous at a point if the limit of the function as the input approaches that point either does not exist or does not equal the function’s value at that point. A function that is discontinuous at one point may be continuous elsewhere.

In real analysis, several distinct types of discontinuities are commonly distinguished. A removable discontinuity occurs when the limit exists but is not equal to the function’s value at that point, and redefining the value at the point would make the function continuous. A jump discontinuity (also called a discontinuity of the first kind) happens when the left-hand and right-hand limits exist but are not equal, producing a sudden jump in the function’s value. An infinite or essential discontinuity occurs when the function grows without bound near the point, so the limit diverges. A second-kind or oscillatory discontinuity is one where the limit does not exist due to persistent oscillation near the point.

Discontinuities also appear in other contexts. In signal processing, a discontinuity is a sudden change in a signal, such as a step, impulse, or spike. In topology and analysis, the concept relates to how a function behaves with respect to limits and neighborhoods, with continuous maps preserving limit behavior. Examples often cited include the Heaviside step function, which has a jump discontinuity at zero, and f(x) = 1/x, which is discontinuous at zero.

Overall, discontinuity is a central concept for understanding where and how mathematical objects fail to be smooth or predictable, guiding the study of limits, differentiation, and approximation.