uniform convergence of two variable-functions

Question

Let $f(x,y)$ be a real-valued two-variable function on the plane.

(1). $\lim_{x\to\infty,y\to\infty}f(x,y)$ exists.

(2). for any fixed $y$, $\lim_{x\to\infty}f(x,y)$ exists and converges uniformly for all $y\in \mathbb{R}$.

(3). for any fixed $x$, $\lim_{y\to\infty}f(x,y)$ exists and converges uniformly for all $x\in \mathbb{R}$.

Question: Whether does (1) imply (2) and (3) or not? I want to find a sufficient condition of (2) and (3).