Punktweise

Punktweise is a German mathematical term meaning pointwise. It describes properties or operations that apply to each point of a domain individually, rather than in an overall or uniform manner.

In analysis, punktweise Konvergenz (pointwise convergence) refers to a mode of convergence for a sequence of functions f_n: X -> Y to a function f: X -> Y, such that for every x in X, the limit lim_{n→∞} f_n(x) = f(x). This is contrasted with uniform convergence, where convergence occurs uniformly in x; uniform convergence implies pointwise convergence, but not vice versa.

A common example is f_n(x) = x^n on the interval [0,1]. For each x in [0,1), f_n(x) → 0, while f_n(1) = 1 for all n, so the pointwise limit is the function f(x) = 0 for x in [0,1) and f(1) = 1. This shows that pointwise limits can be discontinuous even if all f_n are continuous.

Pointwise convergence has different analytic consequences from uniform convergence. For instance, the limit of continuous functions under pointwise convergence need not be continuous, whereas uniform convergence preserves continuity. In measure theory and integration, pointwise convergence is a common starting point, with additional theorems (such as the dominated or monotone convergence theorems) addressing limits of integrals under further hypotheses.

Besides convergence, punktweise is also used to describe a function defined by a rule at each point of the domain (as opposed to a single global formula). In practice, the related term piecewise, or stückweise, is often used for functions defined by different expressions on subdomains.