Theorem relating continuity to graphs
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.
A blog post by T. Tao lists several closed graph theorems throughout mathematics.
Graphs and maps with closed graphs.
If formula_1 is a map between topological spaces then the graph of formula_2 is the set formula_3 or equivalently,
formula_4
It is said that the graph of formula_2 is closed if formula_6 is a closed subset of formula_7 (with the product topology).
Any continuous function into a Hausdorff space has a closed graph (see )
Any linear map, formula_8 between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) formula_9 is sequentially continuous in the sense of the product topology, then the map formula_9 is continuous and its graph, Gr "L", is necessarily closed. Conversely, if formula_9 is such a linear map with, in place of (1a), the graph of formula_9 is (1b) known to be closed in the Cartesian product space formula_7, then formula_9 is continuous and therefore necessarily sequentially continuous. 
Examples of continuous maps that do "not" have a closed graph.
If formula_15 is any space then the identity map formula_16 is continuous but its graph, which is the diagonal formula_17, is closed in formula_18 if and only if formula_15 is Hausdorff. In particular, if formula_15 is not Hausdorff then formula_16 is continuous but does not have a closed graph. 
Let formula_15 denote the real numbers formula_23 with the usual Euclidean topology and let formula_24 denote formula_23 with the indiscrete topology (where note that formula_24 is not Hausdorff and that every function valued in formula_24 is continuous). Let formula_1 be defined by formula_29 and formula_30 for all formula_31. Then formula_1 is continuous but its graph is not closed in formula_7.
Closed graph theorem in point-set topology.
In point-set topology, the closed graph theorem states the following:
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Closed graph theorem —  If formula_1 is a map from a topological space formula_15 into a Hausdorff space formula_36 then the graph of formula_2 is closed if formula_1 is continuous. The converse is true when formula_24 is compact. (Note that compactness and Hausdorffness do not imply each other.)
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First part: just note that the graph of formula_2 is the same as the pre-image formula_41 where formula_42 is the diagonal in formula_43.
Second part: 
For any open formula_44 , we check formula_45 is open. So take any formula_46 , we construct some open neighborhood formula_47 of formula_48 , such that formula_49 .
Since the graph of formula_2 is closed, for every point formula_51 on the "vertical line at x", with formula_52 , draw an open rectangle formula_53 disjoint from the graph of formula_2 . These open rectangles, when projected to the y-axis, cover the y-axis except at formula_55 , so add one more set formula_56.
Naively attempting to take formula_57 would construct a set containing formula_48, but it is not guaranteed to be open, so we use compactness here.
Since formula_24 is compact, we can take a finite open covering of formula_24 as formula_61.
Now take formula_62. It is an open neighborhood of formula_48, since it is merely a finite intersection. We claim this is the open neighborhood of formula_47 that we want.
Suppose not, then there is some unruly formula_65 such that formula_66 , then that would imply formula_67 for some formula_68 by open covering, but then formula_69 , a contradiction since it is supposed to be disjoint from the graph of formula_2 .
If "X", "Y" are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see .
Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact formula_24 is the real line, which allows the discontinuous function with closed graph formula_72.
Also, closed linear operators in functional analysis (linear operators with closed graphs) are typically not continuous.
For set-valued functions.
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Closed graph theorem for set-valued functions —  For a Hausdorff compact range space formula_24, a set-valued function formula_74 has a closed graph if and only if it is upper hemicontinuous and "F"("x") is a closed set for all formula_75.
In functional analysis.
If formula_76 is a linear operator between topological vector spaces (TVSs) then we say that formula_77 is a closed operator if the graph of formula_77 is closed in formula_7 when formula_7 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.
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Theorem — 
A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed.
The theorem is a consequence of the open mapping theorem; see below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).
Relation to the open mapping theorem.
Often, the closed graph theorems are obtained as corollaries of the open mapping theorems in the following way. Let formula_1 be any map. Then it factors as
formula_82.
Now, formula_68 is the inverse of the projection formula_84. So, if the open mapping theorem holds for formula_85; i.e., formula_85 is an open mapping, then formula_68 is continuous and then formula_2 is continuous (as the composition of continuous maps).
For example, the above argument applies if formula_2 is a linear operator between Banach spaces with closed graph, or if formula_2 is a map with closed graph between compact Hausdorff spaces.
Notes.
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References.
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