I'm wondering about the following situation. Suppose $f\colon\mathbb{R}^2\to \mathbb{R}$ is a function. If $f$ is continuous at $(x_0,y_0)$, why does $$ \lim_{x\to x_0}\limsup_{y\to y_0}f(x,y)=f(x_0,y_0)=\lim_{y\to y_0}\limsup_{x\to x_0}f(x,y) $$
and similarly if we take $\liminf$ instead of $\limsup$? Here I'm using the definition $$ \limsup_{y\to y_0}f(y)=\inf_{\epsilon>0}\sup_{y:|y-y_0|\leq\epsilon}f(y). $$
I'm unsure how to compute $\limsup_{y\to y_0}f(x,y)$ since I've only dealt with limit superiors of sequences, and limits of single variable functions. My guess is $\limsup_{y\to y_0}f(x,y)=f(x,y_0)$. After that, since $f$ is continuous at $(x_0,y_0)$, it is continuous at each coordinate, so $f(x,y_0)$ should be continuous at $x_0$, so $\lim_{x\to x_0}f(x,y_0)=f(x_0,y_0)$.
I'm interested in seeing how you could rigorously compute $\limsup_{y\to y_0}f(x,y)$. I think I could try the other three situations once I see that. Thanks!