Exchange of two limits of a function

real-analysis

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?.

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