Pointwise

Pointwise refers to a notion defined at each individual point of a domain, as opposed to globally or uniformly across the domain. In analysis, the phrase is most often used with sequences of functions and with limits taken at each point separately.

Definition and contrast: Let D be a set and f_n: D → R (or into another metric space). We say f_n converges pointwise to a function f on D if, for every x in D, the sequence f_n(x) converges to f(x) as n → ∞. The convergence is judged point by point. By contrast, uniform convergence requires that the convergence happens uniformly for all x in D: for every ε > 0 there exists N such that for all n ≥ N and all x in D, |f_n(x) − f(x)| < ε. Uniform convergence imposes stronger global control over the functions.

Examples: Consider f_n(x) = x^n on the domain [0, 1]. This sequence converges pointwise to a function f, where f(x) = 0 for 0 ≤ x < 1 and f(1) = 1. The convergence is not uniform on [0, 1], though on any interval [0, a] with a < 1 it is uniform. Another common example is a sequence of functions that converges to a limit at each fixed x but with the rate depending on x.

Related concepts: Pointwise convergence extends to sequences of random variables, where one can distinguish between pointwise convergence and convergence almost surely. In algebra and analysis, many operations are defined pointwise, such as (f · g)(x) = f(x)g(x). A key consequence in analysis is that pointwise convergence alone does not preserve properties like continuity, unlike uniform convergence, which does under suitable conditions.