Pointwise convergence

A notion of convergence in mathematics

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.^[1][2]

Definition

The pointwise limit of continuous functions does not have to be continuous: the continuous functions ${\displaystyle \sin ^{n}(x)}$ (marked in green) converge pointwise to a discontinuous function (marked in red).

Suppose that ${\displaystyle X}$ is a set and ${\displaystyle Y}$ is a topological space, such as the real or complex numbers or a metric space, for example. A sequence of functions ${\displaystyle \left(f_{n}\right)}$ all having the same domain ${\displaystyle X}$ and codomain ${\displaystyle Y}$ is said to converge pointwise to a given function ${\displaystyle f:X\to Y}$ often written as $${\displaystyle \lim _{n\to \infty }f_{n}=f\ {\mbox{pointwise}}}$$if (and only if) the limit of the sequence ${\displaystyle f_{n}(x)}$ evaluated at each point ${\displaystyle x}$ in the domain of ${\displaystyle f}$ is equal to ${\displaystyle f(x)}$, written as $${\displaystyle \forall x\in X.\lim _{n\to \infty }f_{n}(x)=f(x).}$$The function ${\displaystyle f}$ is said to be the pointwise limit function of the ${\displaystyle \left(f_{n}\right).}$

The definition easily generalizes from sequences to nets ${\displaystyle f_{\bullet }=\left(f_{a}\right)_{a\in A}}$. We say ${\displaystyle f_{\bullet }}$ converge pointwises to ${\displaystyle f}$, written as $${\displaystyle \lim _{a\in A}f_{a}=f\ {\mbox{pointwise}}}$$if (and only if) ${\displaystyle f(x)}$ is the unique accumulation point of the net ${\displaystyle f_{\bullet }(x)}$ evaluated at each point ${\displaystyle x}$ in the domain of ${\displaystyle f}$, written as $${\displaystyle \forall x\in X.\lim _{a\in A}f_{a}(x)=f(x).}$$

Sometimes, authors use the term bounded pointwise convergence when there is a constant ${\displaystyle C}$ such that ${\displaystyle \forall n,x,\;|f_{n}(x)|<C}$ .^[3]

Properties

This concept is often contrasted with uniform convergence. To say that $${\displaystyle \lim _{n\to \infty }f_{n}=f\ {\mbox{uniformly}}}$$means that $${\displaystyle \lim _{n\to \infty }\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in A\,\}=0,}$$where ${\displaystyle A}$ is the common domain of ${\displaystyle f}$ and ${\displaystyle f_{n}}$, and ${\displaystyle \sup }$ stands for the supremum. That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent. For example, if ${\displaystyle f_{n}:[0,1)\to [0,1)}$ is a sequence of functions defined by ${\displaystyle f_{n}(x)=x^{n},}$ then ${\displaystyle \lim _{n\to \infty }f_{n}(x)=0}$ pointwise on the interval ${\displaystyle [0,1),}$ but not uniformly.

The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, $${\displaystyle f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}}$$