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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
  • 8
    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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