In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A real function ${\displaystyle y=f(x)}$ is closed if the graph is closed, meaning that it contains all of its limit points. Every such continuous function has a closed graph, but the converse is not necessarily true.

More generally, a function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y.

This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.

We give the more general definition of when a Y-valued function or set-valued function defined on a subset S of X has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace S of a topological vector space X (and not necessarily defined on all of X). This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.

When reading literature in functional analysis, if f : X → Y is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then "f is closed" will almost always means the following:

Otherwise, especially in literature about point-set topology, "f is closed" may instead mean the following:

These two definitions of "closed map" are not equivalent. If it is unclear, then it is recommended that a reader check how "closed map" is defined by the literature they are reading.

Throughout, let X and Y be topological spaces.