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Prove or disprove: let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a mapping with the following properties: for each $y \in \mathbb{R}$ the function $x\mapsto f\left(x,y\right)$ is continuous on $\mathbb{R}$, and for each $x\in\mathbb{R}$ the function $y\mapsto f\left(x,y\right)$ is continuous on $\mathbb{R}$. Then $f$ is continuous on $\mathbb{R}^{2}$.

My intuition says it's not true, but I can't think of a simple counterexample.

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if you add the condition that $f$ sends compact sets to compact sets then you have your conclusion – clark Feb 11 '13 at 14:40
up vote 5 down vote accepted

It is not true. See $\S18$ Exercise $12$ from Munkres' book.

enter image description here

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Another reference: this is Example 9.1 in Gelbaum and Olmstead's Counterexamples in Analysis. – David Mitra Feb 11 '13 at 14:36
Note that $F$ is given in polar coordinates by $F(r,\theta) = \frac{\sin(2\theta)}{2}$ which sheds some light on the geometry of the counterexample and from where it came from. – levap Feb 11 '13 at 14:39

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