Characterize a continuous function in terms of its graph (I believe) a map $\varphi: G_1 \to G_2$ is a group homomorphism iff $\operatorname{Graph}{(\varphi)}$ is a subgroup of $G_1 \times G_2$ (similarly for the categories of vector spaces, algebras, rings).
Is there a characterization of a continuous map $f: X \to Y$ between topological spaces in terms of its graph? Or perhaps for spaces with more structure, such as manifolds, or smooth manifolds?
Edit: The closed graph theorem provides some answer. 
 A: For general maps between general topological spaces it is surprisingly easy to show that continuous maps cannot be characterized by topological properties of 
their graphs. The reason is that a continuous bijection between homeomorphic
spaces is not necessarily a homeomorphism. (In contrast: a bijective group
homomorphism is always an isomorphism, i.e. its inverse is also a homomorphism) 
To show that a characterization does not exist, let $f: X \to X$ be a 
continuous bijection with a discontinuous inverse. Then the graphs of $f$ and
$f^{-1}$ are both subsets of $X \times X$, and the map $(x, y) \mapsto (y, x)$
is a homeomorphism $X \times X \to X \times X$ which exchanges the two graphs.
Clearly neither graph has any topological property, either intrinsic or in
relation to the ambient topology, that the other graph does not have.
There are of course classes of maps where this is not an obstruction.
Continuous bijections between compact Hausdorff spaces are always homeomorphisms, and the same is true for connected
linear order spaces. A popular subclass of the latter is formed by 
functions $\mathbb{R} \to \mathbb{R}$. Elementary Topology; Textbook in Problems has some interesting criteria for those in exercises
19.33-19.36. (note also exercises 19.27 and 19.28 where a stronger closed
graph theorem is given than that in Wikipedia)
The case of compact manifolds is covered by the closed graph theorem you referenced. Whether something similar applies to non-compact manifolds I
don't know.
