MathDB

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

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}.

B4