TITLE: Characterizing continuous functions based on the graph of the function
QUESTION [8 upvotes]: I had asked this question: Characterising Continuous functions some time back, and this question is more or less related to that question.
Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ and suppose the set  $G = \\{ (x,f(x) : x \in \mathbb{R}\\}$ is connected and closed in $\mathbb{R}^{2}$, then does it imply $f$ is continuous?

REPLY [14 votes]: This Monthly paper has short simple proofs of the following
THEOREM$\ $ TFAE if $\rm\ f: \mathbb R\to \mathbb R\ $ has a closed graph in $\:\mathbb R^2$
(a)$\rm\ \ f\ $ is continuous.
(b)$\rm\ \  f\ $ is locally bounded.
(c)$\rm\ \ f\ $ has the intermediate value property.
(d)$\rm\ \ f\ $ has a connected graph in $\rm\mathbb R^2$.
More generally the result is merely a special case of R. L. Moore's 1920 characterization of a topological line as a locally compact metric space that is separated into two connected sets by each of its points.
Per request, I've appended the proof of the theorem below.