Does $\lim_{x\to a}\lim_{y\to b}f(x,y)$ allways equal $\lim_{y\to b}\lim_{x\to a}f(x,y)$?

Question

Does $\lim_{x\to a}\lim_{y\to b}f(x,y)$ allways equal $\lim_{y\to b}\lim_{x\to a}f(x,y)$? Obviously it does if $f$ is continues, but I couldnt think of a counterexample with a non continues function.

Answer 1

What about $$ \lim_{y\to 0}\lim_{x\to 0}\frac{1}{1+(x/y)^2} = 1$$ in comparison to $$\lim_{x\to 0}\lim_{y\to 0} \frac{1}{1+(x/y)^2} = 0\;?$$

Answer 2

Consider for example the function $f:\mathbb R^2\to\mathbb R$ defined by $$f(x, y)=\begin{cases} y/x&\text{if }x\neq 0\\ 0&\text{if } x= 0\end{cases}$$ We have $$\lim_{x\to 0}\,\lim_{y\to 0}\,f(x,y)= 0$$ but $$\lim_{y\to 0}\,\lim_{x\to 0}\,f(x,y)$$ doesn't even exist. The problem here is that the limit $\lim_{y\to 0} f(x, y)$ is not uniform on $x$ (for $x$ close to $0$, we need smaller $y$ in order to guarantee $|f(x, y|<\epsilon$). Under the assumption of uniform limit, we can actually have the equality (Moore-Osgood theorem).