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Intuitively, a function $\mathbb{R}\rightarrow\mathbb{R}$ is continuous if you can draw its graph without taking the pen off the page. This suggests the following theorem:

A map $f:X \rightarrow Y$ is continuous if and only if $f$ is connected in the product topology $X \times Y$.

Is this true? And if not, can anyone think of an additional premise or two that would make it true?

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Not if $X$ is disconnected. – Thomas Andrews Aug 17 '12 at 18:56
The Topologists Sine Curve is not continuous at $x=0$. – David Mitra Aug 17 '12 at 18:57
I'd be curious if you add the condition that $X$ is path-connected, then is it true that $f$ is continuous if and only if the graph of $f$ is path-connected. – Thomas Andrews Aug 17 '12 at 19:01
@Thomas: not necessarily. Slice the plane along a half-line and bend one side of the half-line down a little and the other one up a little. You get a discontinuous function $\mathbb{R}^2 \to \mathbb{R}$ with path-connected graph. – t.b. Aug 17 '12 at 19:07

It isn't true in general. An obvious variant of the Topologist's sine curve provides an example of a function $f:\Bbb R\rightarrow \Bbb R$ whose graph is connected but fails to be continuous (at $x=0$).

However, this article shows that "it is correct to conclude that continuous real functions over $\Bbb R$ are those functions over $\Bbb R$ whose graphs, in the plane $\Bbb R^2$, are both closed and connected".

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Does anyone know of a non-jstor link to the article I mentioned? I can only view the first page. – David Mitra Aug 17 '12 at 19:30
I don't think the Monthly is freely available anywhere. It would be a bit better (more reliable) to use the stable link to JSTOR: than the one you use. $$ $$ In his Real analysis, Carothers mentions Burgess's paper you link to and also refers to Boas's A primer of real functions and Randolph's Basic real and abstract analysis for that result (he doesn't say where to find it in those books). – t.b. Aug 17 '12 at 19:49
@DavidMitra In my profile you can find my email address. I have access to the paper and I can send you it by email. If you are asking not just because you want a copy but because you wanted to provide a link for other users, if there is a freely accessible link, I did not found such a link. – Martin Sleziak Aug 18 '12 at 6:58
The following result seems to be related: Theorem 8.2 in van Rooij-Schikhof, p.52 For a function $f \colon [a,b]\to\mathbb R$ the following are equivalent: $f$ is continuous. $\Leftrightarrow$ $\Gamma_f$ is compact. $\Leftrightarrow$ $\Gamma_f$ is arcwise connected. (Here $\Gamma_f$ denotes the graph of $f$.) – Martin Sleziak Aug 18 '12 at 9:42

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