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.

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 ${\displaystyle f_{\bullet }}$ converges pointwise 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 ${\displaystyle \forall n,x,\;|f_{n}(x)|<C}$ .

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, $${\displaystyle f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}}$$ takes the value ${\displaystyle 1}$ when ${\displaystyle x}$ is an integer and ${\displaystyle 0}$ when ${\displaystyle x}$ is not an integer, and so is discontinuous at every integer.

The values of the functions ${\displaystyle f_{n}}$ need not be real numbers, but may be in any topological space, in order that the concept of pointwise convergence make sense. Uniform convergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in metric spaces, and, more generally, in uniform spaces.

Let ${\displaystyle Y^{X}}$ denote the set of all functions from some given set ${\displaystyle X}$ into some topological space ${\displaystyle Y.}$ As described in the article on characterizations of the category of topological spaces, if certain conditions are met then it is possible to define a unique topology on a set in terms of which nets do and do not converge. The definition of pointwise convergence meets these conditions and so it induces a topology, called the topology of pointwise convergence, on the set ${\displaystyle Y^{X}}$ of all functions of the form ${\displaystyle X\to Y.}$ A net in ${\displaystyle Y^{X}}$ converges in this topology if and only if it converges pointwise.