Putnam 1974 B4

A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be continuous in each variable separately if, for each fixed value $y_0$ of $y$ , the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$ . Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that $$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$ for all $(x,y)\in \mathbb{R}^{2}.$