The title pretty much says it all. For some $f(x,y)$, in what cases is it true that
$$ \lim_{x \to a} \lim_{y \to b} f(x,y)=\lim_{y \to b} \lim_{x \to a} f(x,y) $$
I was able to find this, but it doesn't quite answer what I'm asking.
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try searching for uniform continuity – Jan Aug 24 '15 at 00:29
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Thanks for the pointer, but I'm confused as to the connection. Additionally, what if $f(x,y)$ is a discrete function and/or the limits are evaluated at infinity? – crb233 Aug 24 '15 at 00:46
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This question comes up in a lot of different guises in various places in analysis, and frankly there is no good general answer, to my knowledge. The "advanced calculus criterion" is that the $y$ limit converges uniformly in $x$ or vice versa. But this is unacceptably rigid in a lot of applications (e.g. interchange of limit and integral). – Ian Aug 24 '15 at 01:12
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@CurtisBechtel For sequeneses you can see the previous answer. Otherwise can look for Moore-Osgood theorem. – A.Γ. Aug 24 '15 at 01:49
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It's true if $f$ is continuous in a neighbourhood of $(a,b)$, both limits being $f(a,b)$.
Robert Israel
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Is this valid for limits at infinity if the function is shown to be continuous? – crb233 Aug 24 '15 at 01:11
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@CurtisBechtel, yes. Consider the function $f(1/x, 1/y)$ near $(0, 0)$. – vonbrand Aug 24 '15 at 01:34
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If the function is continuous in a neighbourhood of infinity (in the one-point compactification of the plane). – Robert Israel Aug 24 '15 at 03:11