The graph of the cubic function ${\displaystyle f(x)=x^{3}-9x}$ on the interval ${\displaystyle [-4,4]}$ is closed because the function is continuous. The graph of the Heaviside function on ${\displaystyle [-2,2]}$ is not closed, because the function is not continuous.

In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.

Graphs and maps with closed graphs

If ${\displaystyle f:X\to Y}$ is a map between topological spaces then the graph of ${\displaystyle f}$ is the set ${\displaystyle \operatorname {Gr} f:=\{(x,f(x)):x\in X\))$ or equivalently,

$${\displaystyle \operatorname {Gr} f:=\{(x,y)\in X\times Y:y=f(x)\))$$

the graph of ${\displaystyle f}$ is closed

${\displaystyle \operatorname {Gr} f}$

${\displaystyle X\times Y}$

Any continuous function into a Hausdorff space has a closed graph.

Any linear map, ${\displaystyle L:X\to Y,}$ between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) ${\displaystyle L}$ is sequentially continuous in the sense of the product topology, then the map ${\displaystyle L}$ is continuous and its graph, Gr L, is necessarily closed. Conversely, if ${\displaystyle L}$ is such a linear map with, in place of (1a), the graph of ${\displaystyle L}$ is (1b) known to be closed in the Cartesian product space ${\displaystyle X\times Y}$, then ${\displaystyle L}$ is continuous and therefore necessarily sequentially continuous.^[1]

Examples of continuous maps that do not have a closed graph

If ${\displaystyle X}$ is any space then the identity map ${\displaystyle \operatorname {Id} :X\to X}$ is continuous but its graph, which is the diagonal ${\displaystyle \operatorname {Gr} \operatorname {Id} :=\{(x,x):x\in X\},}$, is closed in ${\displaystyle X\times X}$ if and only if ${\displaystyle X}$ is Hausdorff.^[2] In particular, if ${\displaystyle X}$ is not Hausdorff then ${\displaystyle \operatorname {Id} :X\to X}$ is continuous but does not have a closed graph.

Let ${\displaystyle X}$ denote the real numbers ${\displaystyle \mathbb {R} }$ with the usual Euclidean topology and let ${\displaystyle Y}$ denote ${\displaystyle \mathbb {R} }$ with the indiscrete topology (where note that ${\displaystyle Y}$ is not Hausdorff and that every function valued in ${\displaystyle Y}$ is continuous). Let ${\displaystyle f:X\to Y}$ be defined by ${\displaystyle f(0)=1}$ and ${\displaystyle f(x)=0}$ for all ${\displaystyle x\neq 0}$. Then ${\displaystyle f:X\to Y}$ is continuous but its graph is not closed in ${\displaystyle X\times Y}$.^[3]

Closed graph theorem in point-set topology

In point-set topology, the closed graph theorem states the following:

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact ${\displaystyle Y}$ is the real line, which allows the discontinuous function with closed graph ${\displaystyle f(x)={\begin{cases}{\frac {1}{x)){\text{ if ))x\neq 0,\\0{\text{ else))\end{cases))}$.

For set-valued functions

In functional analysis

If ${\displaystyle T:X\to Y}$ is a linear operator between topological vector spaces (TVSs) then we say that ${\displaystyle T}$ is a closed operator if the graph of ${\displaystyle T}$ is closed in ${\displaystyle X\times Y}$ when ${\displaystyle X\times Y}$ is endowed with the product topology.

The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. The original result has been generalized many times. A well known version of the closed graph theorems is the following.