Inmathematics,pointwise convergence is one ofvarious senses in which asequence offunctions canconverge to a particular function. It is weaker thanuniform convergence, to which it is often compared.^[1][2]

Suppose that${\displaystyle X}$ is a set and${\displaystyle Y}$ is atopological space, such as thereal orcomplex numbers or ametric space, for example. Asequence offunctions${\displaystyle \left(f_n\right)}$ all having the same domain${\displaystyle X}$ andcodomain${\displaystyle Y}$ is said toconverge pointwise to a given function${\displaystyle f:X\to Y}$ often written as$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{f}_n=f\text{pointwise}}$$if (and only if) thelimit 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$${\displaystyle \mathrm{\forall }x\in X,\underset{n\to \mathrm{\infty }}{lim}{f}_n(x)=f(x).}$$The function${\displaystyle f}$ is said to be thepointwise limit function of the${\displaystyle \left(f_n\right).}$

The definition easily generalizes from sequences tonets${\displaystyle f_{\bullet }={\left(f_a\right)}_{a\in A}}$. We say${\displaystyle f_{\bullet }}$ converges pointwise to${\displaystyle f}$, written as$${\displaystyle \underset{a\in A}{lim}{f}_a=f\text{pointwise}}$$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$${\displaystyle \mathrm{\forall }x\in X,\underset{a\in A}{lim}{f}_a(x)=f(x).}$$

Sometimes, authors use the termbounded pointwise convergence when there is a constant${\displaystyle C}$ such that .^[3]

This concept is often contrasted withuniform convergence. To say that$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{f}_n=f\text{uniformly}}$$means that$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}sup\{\left|f_n(x)-f(x)\right|:x\in A\}=0,}$$where${\displaystyle A}$ is the common domain of${\displaystyle f}$ and${\displaystyle f_n}$, and${\displaystyle sup}$ stands for thesupremum. 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${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{f}_n(x)=0}$ pointwise on the interval${\displaystyle [0,1),}$ but not uniformly.

The pointwise limit of a sequence ofcontinuous functions may be a discontinuous function, but only if the convergence is not uniform. For example,$${\displaystyle f(x)=\underset{n\to \mathrm{\infty }}{lim}\cos (\pi x{)}^{2n}}$$takes the value${\displaystyle 1}$ when${\displaystyle x}$ is aninteger 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 anytopological 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 inmetric spaces, and, more generally, inuniform spaces.

Let${\displaystyle Y^X}$ denote the set of all functions from some given set${\displaystyle X}$ into sometopological space${\displaystyle Y.}$As described in the article oncharacterizations 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 whichnets do and do notconverge. The definition of pointwise convergence meets these conditions and so it induces atopology, calledthetopology 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.

The topology of pointwise convergence is the same as convergence in theproduct topology on the space${\displaystyle Y^X,}$ where${\displaystyle X}$ is the domain and${\displaystyle Y}$ is the codomain. Explicitly, if${\displaystyle \mathcal{F}\subseteq Y^X}$ is a set of functions from some set${\displaystyle X}$ into some topological space${\displaystyle Y}$ then the topology of pointwise convergence on${\displaystyle \mathcal{F}}$ is equal to thesubspace topology that it inherits from theproduct space${\displaystyle \prod _{x\in X}Y}$ when${\displaystyle \mathcal{F}}$ is identified as a subset of this Cartesian product via the canonical inclusion map${\displaystyle \mathcal{F}\to \prod _{x\in X}Y}$ defined by${\displaystyle f\mapsto (f(x){)}_{x\in X}.}$

If the codomain${\displaystyle Y}$ iscompact, then byTychonoff's theorem, the space${\displaystyle Y^X}$ is also compact.

Inmeasure theory, one talks aboutalmost everywhere convergence of a sequence ofmeasurable functions defined on ameasurable space. That means pointwise convergencealmost 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 atopology on the space of measurable functions on ameasure space (although it is aconvergence structure). For in a topological space, when every subsequence of a sequence has itself a subsequence with the samesubsequential limit, the sequence itself must converge to that limit.

But consider the sequence of so-called "galloping rectangles" functions (also known as the typewriter sequence), which are defined using thefloor function: let${\displaystyle N=\mathrm{floor}\left({\log }_{2}n\right)}$ and${\displaystyle k=n}$mod${\displaystyle 2^N,}$ and let

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

• Box topology – Mathematical concept
• Convergence space – Generalization of the notion of convergence that is found in general topology
• Cylinder set – Natural basic set in product spaces
• List of topologies – List of concrete topologies and topological spaces
• Modes of convergence (annotated index) – Annotated index of various modes of convergence
• Strong operator topology – Locally convex topology on function spaces
• Topologies on spaces of linear maps
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• Convergence (mathematics)
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