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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.2012-08-17
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    The [Topologists Sine Curve](http://en.wikipedia.org/wiki/Topologist%27s_sine_curve) is not continuous at $x=0$.2012-08-17
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    http://math.stackexchange.com/questions/34103/for-a-function-from-mathbbr-to-itself-whose-graph-is-connected-in-mathbb2012-08-17
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    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.2012-08-17
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    @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.2012-08-17
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    The 19th-century intuition of a continuous function was exactly what you describe. But in that intuition, the function defined by [$f(x) = 0$ for $x\le 0$, and $f(x)=1$ for $x\gt 0$] could be continuous, since you can draw its graph without taking the pen off the paper. You have to move the pen vertically when you get to $x=0$, but you do not have to take it off the paper, and in the 19th century, this function was considered continuous in some contexts. So it is not just being pernickety to say that your intuitive description does not accord with the modern notion of continuity.2012-08-17
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    (I believe my source for this somewhat strange-seeming claim is Judith Grabiner's *The Origins of Cauchy's Rigorous Calculus*. I will try to find a suitable quotation.)2012-08-17
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    The example of @t.b. can be thought of as the map $\mathrm{Arg} : \mathbb{C}\setminus \{ 0 \} \to\mathbb{R}$ which associates to a complex number the _principal value_ of the argument of that number. Its graph is a piece of a [helicoid](http://en.wikipedia.org/wiki/Helicoid). In programming, that map is often called [`atan2`](http://en.wikipedia.org/wiki/Atan2) (link contains illustration).2014-08-06

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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.2012-08-17
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    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: http://www.jstor.org/stable/2324521 than the one you use. $$ $$ In his *[Real analysis](http://books.google.com/books?id=4VFDVy1NFiAC)*, Carothers mentions Burgess's paper you link to and also refers to Boas's *[A primer of real functions](http://books.google.com/books?id=uQtykVwbrm4C)* and Randolph's *[Basic real and abstract analysis](http://books.google.com/books?id=NRrvAAAAMAAJ)* for that result (he doesn't say where to find it in those books).2012-08-17
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    @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.2012-08-18
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    The following result seems to be related: Theorem 8.2 in van Rooij-Schikhof, [p.52](http://books.google.com/books?id=Cqk5AAAAIAAJ&pg=PA52) 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 $f$.)2012-08-18