When one studies multi-variable functions, usually starting with functions from $\mathbb R^2$ to $\mathbb R$, at the very beginning we are taught that iterating limits and "normal limits" (i.e by definition) does not behave equally, and that the existance of one does not promise the existence of the other, for example:
$$f(x,y) = \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\ne(0,0) \\ 0 & (x,y) = (0,0)\end{cases}$$
for this function we get that: $$\lim_{x\to 0}\lim_{y\to 0}= \lim_{y\to 0}\lim_{x\to 0} = 0$$ but this limit does not exist: $$\lim_{(x,y)\to (0,0)}f(x,y)$$ $$$$ Or the other way, with this function serving as an example:
$$f(x,y) = \begin{cases} x+y\sin(\frac{1}{x}) & x\ne0\\0&x=0 \end{cases}$$
here we get that: $$\lim_{(x,y)\to (0,0)}f(x,y)=0$$
but this limit does not exist: $$\lim_{y\to0}\lim_{x\to0} f(x,y)$$
$$$$ My question is this: Is there, under any presumtions or conditions, any connection between the "normal limit" and the iterated limits, which makes them "iff" statements, or that if one exists, the other also exists (and the other way as well). If so, what are they, and how do we prove them?
(I will note that this is not the same question that can be found here: Limits of 2 variable functions, or here: Iterated Limits, since my question does not include any assumption that the all the limits exist and (some of them) are equal, and in fact, I ask if any conditions or presumptions hold (maybe for a group of functions?) we can get the connection).
Thanks!
Edit: Is the "Moore-Osgood theorem" relevant for this? Wikipedia states that "If ${\displaystyle \lim _{x\to p}f(x,y)}$ exists pointwise for each $y$ different of $q$ and if ${\displaystyle \lim _{y\to q}f(x,y)}$ converges uniformly for $x≠p$ then the double limit and the iterated limits exist and are equal.", also found in this link: http://www.math.unm.edu/~loring/links/analysis_f10/exchange.pdf . If so, I didn't understand it, and would love your explanation on it.
