kontinuierliche

Kontinuierliche is a German term meaning continuous. In mathematical and scientific contexts it is used to describe things that are unbroken, with no gaps or jumps, and it appears in phrases such as kontin­uierliche Funktion (continuous function) or kontinuierlicher Prozess (continuous process). The word is applied across disciplines to denote a steady, uninterrupted progression or state, as opposed to discrete or intermittent ones.

In mathematics, kontinuiertheit is central to the study of change and structure. A function f is continuous at a point x0 if the limit of f as x approaches x0 equals f(x0). A function is continuous on a set if it is continuous at every point of that set. In topology, continuity is defined via the behavior of preimages of open sets under a function. Common examples of continuous functions on the real numbers include polynomials, exponentials, and trigonometric functions; discontinuous examples include piecewise or step functions like the floor function, which have jumps at certain points.

The concept underpins real analysis and is closely related to limits and convergence. The development of a precise, epsilon-delta definition of continuity occurred in the 19th century, with key contributions from mathematicians such as Karl Weierstrass. This formalization helped distinguish continuous phenomena from discontinuous ones in both theory and application, including physics, engineering, and computer science.

See also: Continuity, Continuous function, Real analysis, Topology. Notes: in everyday German, kontinuierliche describes a property of smooth, unbroken change; as a stand-alone noun it is uncommon and typically appears within broader phrases.