The following exercise 2.2.9 is borrowed from Terence Tao's Analysis II, page 33. While I understand the problem completely, I lack the technique to attack it. So I will really appreciate a small hint on how to start the proof.
Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function. Let $(x_0 , y_0 ) \in \mathbb{R}^2$ be a point. If $f$ is continuous at $(x_0 , y_0 )$, show that$$\lim_{x\to x_0 } \limsup_{y \to y_0} f (x, y) = \lim_{y\to y_0 } \limsup_{x \to x_0} f (x, y) = f (x_0 , y_0 )$$and$$\lim_{x\to x_0 } \liminf_{y \to y_0} f (x, y) = \lim_{y\to y_0 } \liminf_{x \to x_0} f (x, y) = f (x_0 , y_0 )$$In particular, we have$$\lim_{x\to x_0 } \lim_{y \to y_0} f (x, y) = \lim_{y\to y_0 } \lim_{x \to x_0} f (x, y) = f (x_0 , y_0 )$$whenever the limits on both sides exist.
We have defined $\limsup$ as$$\limsup_{x\to x_0} f(x,y) = \inf \left\{ \sup \{f(x,y): |x-x_0| <r \}: r>0\right\}$$