It is known that a continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $\lim_{x\to\pm\infty}$ exist and are finite is bounded. See, e.g., Prove that a continuous real function with finite limits is bounded
My question: Is there some kind of condition on limits for bivariate functions? I.e., are there conditions for $\lim_{x,y}$, $\lim_x$ and $\lim_y$ such that a continuous $f:\mathbb{R}^2\to\mathbb{R},(x,y)\mapsto f(x,y)$ is bounded $(|f(x,y)|\le M$ for an $M$)?