Closed graph property

A discontinuous function with a closed graph: there are no "missing" (limit) points

Discontinuous function with a graph that is not closed: the point $(0,0)$ is a limit point that is not a point on the graph.

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A real function $y=f(x)$ 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.