Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$

If for all $y\in(y_0-\delta,y_0+\delta)$, $\lim_{x\to x_0}f(x,y)=A$ uniformly (for any $\epsilon>0$ there exists $\eta>0$ such that for any $y\in(y_0-\delta,y_0+\delta)$ and $|x-x_0|<\eta$, we have $|f(x,y)-f(x_0,y_0)|<\epsilon$), can we claim that double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y) $$ exists?

I want some criterions that can be tested...


The definition of $\lim_{(x,y)\to(x_0,y_0)}f(x,y) = L$ is that $f(x,y)$ is defined for points arbitrarily close to $(x_0,y_0)$, and for every $\epsilon > 0$, there is a $\delta^\prime > 0$ such that $\|(x,y)-(x_0,y_0)\| < \delta^\prime$ implies $|f(x,y)-f(x_0,y_0)| < \epsilon$. The condition you state in parentheses implies this: if there is an $\eta > 0$ and a $\delta > 0$ such that $y \in (y_0−\delta,y_0+\delta)$ and $|x−x_0|<\eta$ implies $|f(x,y)−f(x_0,y_0)|< \epsilon$, then you can choose $\delta^\prime$ such that the $\delta^\prime$-ball about $(x_0,y_0)$ is contained in the rectangle $(x_0−\eta,x_0+\eta)\times(y_0−\delta,y_0+\delta)$ (for instance, you can set $\delta^\prime$ to be the minimum of $\delta$ and $\eta$). Then $\|(x,y)-(x_0,y_0)\| < \delta^\prime$ implies $y \in (y_0−\delta,y_0+\delta)$ and $|x−x_0|<\eta$, and hence $|f(x,y)−f(x_0,y_0)|< \epsilon$.

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