There are many posts on this site discussing exchanging the order of a limits of a single variable function. I would like to confirm my application of it to a multivariable function:
If (but not only if) $\lim_{x \to a}f(x,y)$ is defined and continuous for all $y$ in an open interval around $b$ (except possibly at $b$ itself) AND $\lim_{y \to b}f(x,y)$ is defined and continuous for all $x$ in an open interval around $a$ (except possibly at $a$ itself), then $$\lim_{x \to a}[\lim_{y \to b}f(x,y)] = \lim_{y \to b}[\lim_{x \to a}f(x,y)].$$
Is that correct?
