Funktionsgrenze

Funktionsgrenze is a term used in German mathematics to denote the boundary that separates regions related to a function, typically where the function’s domain ends or where its behavior changes. The exact meaning depends on context, but it is commonly associated with the boundary of the domain of a function or with the boundary of a region where the function is nonzero or piecewise defined.

In the context of the domain, the Funktionsgrenze of a function f: D → R (or C) in a larger ambient space X is the topological boundary ∂D. This boundary consists of those points for which every neighborhood intersects both D and its complement. The boundary is important for questions of continuity, extendability, and limits that are taken from outside or toward the edge of the domain.

Other uses occur when a function changes its defining formula or behavior along a dividing set. For example, in piecewise functions the dividing curve or surface can be described as the Funktionsgrenze. The boundary of the function’s support—the region where the function is nonzero—also constitutes a relevant boundary in analysis.

Applications include boundary conditions in partial differential equations and boundary-value problems, where conditions are imposed on the Funktionsgrenze, as well as in complex analysis and potential theory, where boundary behavior influences analytic continuation and function values.

Notes: terminology varies; more common terms are Definitionsbereich (domain), Rand or Grenzbereich (boundary), and Grenzmenge (boundary set).