diskontinuierlichen

Diskontinuierlichen is the inflected form of the German adjective diskontinuierlich, meaning not continuous or having a discontinuity. The form is commonly used in mathematical and scientific contexts to describe objects such as functions, signals, or processes that do not exhibit a continuous behavior. In German texts, diskontinuierlichen appears in phrases like diskontinuierlichen Funktionen or diskontinuierlichen Signalen, typically in plural forms and in various cases after definite determiners.

In mathematics, a function f is discontinuous at a point x0 if the limit of f as x approaches x0 does not equal f(x0) or if the limit does not exist. Discontinuities are classified in several ways. A removable discontinuity occurs when a function can be redefined at x0 to make it continuous. A jump discontinuity arises when the left-hand and right-hand limits exist but are unequal. An essential or non-removable discontinuity occurs when a limit does not exist in a stronger sense. Discontinuities can be isolated, as in piecewise-defined functions, or occur on sets of points with various topological properties.

Beyond pure mathematics, diskontinuierlichen descriptions appear in physics, engineering and signal processing to denote systems or signals that change abruptly rather than smoothly. For example, a piecewise constant signal is diskontinuierlich, featuring abrupt transitions at certain times. In topology and analysis, continuity remains the default assumption, while diskontinuierlichen elements are studied to understand behavior at and near points of discontinuity.

In German grammar, diskontinuierlichen is used as the plural, case-appropriate inflected form of diskontinuierlich, typically following a definite article or determiner.