Closed graph property

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A real function 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.

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