Exchange of two limits of a function

Question

The question is simply given a function $f(x,y)$, when does $\mathop {\lim }\limits_{x \to 0} \mathop {\lim }\limits_{y \to 0} f(x,y) = \mathop {\lim }\limits_{y \to 0} \mathop {\lim }\limits_{x \to 0} f(x,y)$ hold?

I know that if the mixed limit $\mathop {\lim }\limits_{x \to 0,y \to 0} f(x,y)$ exists, then above exchange is valid, but I think there should be more general conditions. Thanks!

Related post about limit of sequence: When can we exchange order of two limits?.

A sufficient condition is that $f$ be separately continuous. — Daniel Fischer, Mar 12 '17 at 14:37

Mark Viola · Accepted Answer · 2017-03-19T15:00:01.063

6

The Moore-Osgood Theorem states that if $\lim_{x\to x_0}f(x,y)$ converges uniformly for $y\ne y_0$ and if $\lim_{y\to y_0}f(x,y)$ converges pointwise for $x\ne x_0$, then the iterated limits are equal with

$$\lim_{x\to x_0}\lim_{y\to y_0}f(x,y)=\lim_{y\to y_0}\lim_{x\to x_0}f(x,y)$$

edited Mar 19 '17 at 15:00

answered Mar 12 '17 at 17:24

Mark Viola

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Do you know where I can find the proof? – Matheus Manzatto Sep 28 '17 at 05:14
@MatheusManzatto I don't have a reference. – Mark Viola Sep 28 '17 at 05:24
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@MatheusManzatto Rudin's Principles of Mathematical Analysis, 3ed, ch. 7. – Teddy Jan 07 '20 at 07:47
Does not it matter which one converges uniformly and which one pointwise? – VIVID Jan 28 '21 at 12:52
@VIVID It doesn't matter. Just interchange $x$ and $y$. – Mark Viola Jan 28 '21 at 15:47
@MarkViola Ok, thank you – VIVID Jan 28 '21 at 15:59
@VIVID You're welcome. My pleasure. – Mark Viola Jan 28 '21 at 16:06
Is the formula still valid when $\lim_{y \to y_0} \lim_{x \to x_0} f(x, y) = \infty$? – Aug 05 '21 at 03:30
@ColinTan If the conditions for the theorem to hold, then the double limit exists and equal to both iterated limits. – Mark Viola Aug 05 '21 at 03:49

Exchange of two limits of a function

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