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Give a proof or counterexample of the following statement: Let $f$ be a real-valued function, that is defined and continuous on all of $\mathbb{R}^2$ except at the origin. It has a removable discontinuity at the origin provided that the limit $\lim_{ (x,y)\to(0,0)} f(x, y)$ exists along all parabolas that contain the origin.

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    Do you have any thoughts on the problem? I would also try to phrase this more as a question than a command. That ruffles some users.2012-02-16
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    It looks like you've transcribed a math problem, but without putting any of your own thoughts or words around it: this is rather off-putting, more in questions written like this than your earlier ones, because people like to know an OP is open for engagement with the community but this sends all the wrong signals.2012-02-16
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    What about $f(x,y)=1$ if $x>0$ and $0, $f(x,y)=0$ otherwise?2012-02-16
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    @JonasMeyer: Your function is not continuous outside the origin.2012-02-16
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    @mrf: Thank you, I completely missed that requirement. I could multiply $f$ by something continuous on $x>0$ that goes from $0$ to $1$ to $0$ on each vertical line segment from $(x,0)$ to $(x,x^3)$, but at that point it is probably getting more complicated than David Mitra's examples.2012-02-17

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