Consider the function $f:\Bbb R^2\to \Bbb R$, defined by $f(x,y)=\begin{cases} \frac{2xy}{x^2+y^2}& (x,y)\neq (0,0\\ 0& (x,y)=(0,0) \end{cases}$ and
$g(x,y)=\sum_{n=1}^\infty \frac{f((x-n),(y-n))}{2^n}$. Which of the following statements are true?
$1$. $g$ is continuous on $\Bbb R^2$.
$2$. $g$ is continuous on $\Bbb R^2\setminus (k,k),k\in\Bbb N$.
$3$. $g(c,y)$ is continuous on $\Bbb R$ for each fixed $c$.
I am trying $M_n$-test for series of $g$ as function $f$ is bounded, but I am confused about series of functions of two variables. I only studied about series of single variable functions. It seems that answer will be options $2$nd and $3$rd but don’t know how . Please help . Thank you in advance.
Hints:
$$g(x,y) = \frac{f(x-1,y-1)}{2} + \sum_{n=2}^\infty \frac{f((x-n),(y-n))}{2^n}.$$
The first term on the right is discontinuous at $(1,1)$ while the remaining sum is continuous at $(1,1)$ by Weierstrass.
Let $E=\mathbb R^2 \setminus \{(n,n):n\in \mathbb N\}.$ Then each summand in the series defining $g$ is continuous everywhere on $E.$ Use Weierstrass.
Note $f$ is continuous on each vertical line. Thus each summand defining $g$ is continuous on each vertical line. Weierstrass then shows $g$ is continuous on each vertical line.