Definition[edit]

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

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

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

Properties[edit]

This concept is often contrasted with uniform convergence. To say that $\lim _{n\to \infty }f_{n}=f\ {\mbox{uniformly}}$ means that $\lim _{n\to \infty }\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in A\,\}=0,$ where $A$ is the common domain of $f$ and $f_{n}$ , and $\sup$ 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 $f_{n}:[0,1)\to [0,1)$ is a sequence of functions defined by $f_{n}(x)=x^{n},$ then $\lim _{n\to \infty }f_{n}(x)=0$ pointwise on the interval $[0,1),$ 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, $f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}$ takes the value $1$ when $x$ is an integer and $0$ when $x$ is not an integer, and so is discontinuous at every integer.

The values of the functions $f_{n}$ 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.

Almost everywhere convergence[edit]

In measure theory, one talks about almost everywhere convergence of a sequence of measurable functions defined on a measurable space. That means pointwise convergence almost everywhere, that is, on a subset of the domain whose complement has measure zero. Egorov's theorem states that pointwise convergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set.

Almost everywhere pointwise convergence on the space of functions on a measure space does not define the structure of a topology on the space of measurable functions on a measure space (although it is a convergence structure). For in a topological space, when every subsequence of a sequence has itself a subsequence with the same subsequential limit, the sequence itself must converge to that limit.

But consider the sequence of so-called "galloping rectangles" functions, which are defined using the floor function: let $N=\operatorname {floor} \left(\log _{2}n\right)$ and $k=n$ mod $2^{N},$ and let $f_{n}(x)={\begin{cases}1,&{\frac {k}{2^{N}}}\leq x\leq {\frac {k+1}{2^{N}}}\\0,&{\text{otherwise}}.\end{cases}}$

Then any subsequence of the sequence $\left(f_{n}\right)_{n}$ has a sub-subsequence which itself converges almost everywhere to zero, for example, the subsequence of functions which do not vanish at $x=0.$ But at no point does the original sequence converge pointwise to zero. Hence, unlike convergence in measure and $L^{p}$ convergence, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.