Answer
No.
Work Step by Step
Consider two different approaches to (0,0):
1. along x=2y, $\lim\limits_{x,y \to 0,0}f(x,y)=\lim\limits_{x,y \to 0,0}\frac{4y^{2}-y^{2}}{4y^{2}+y^{2}}=\frac{3}{5}$
2. along x=3y, $\lim\limits_{x,y \to 0,0}f(x,y)=\lim\limits_{x,y \to 0,0}\frac{9y^{2}-y^{2}}{9y^{2}+y^{2}}=\frac{4}{5}$
Hence, the limit of function $f(x,y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ at (0,0) does not exist,
and we can not define $f(0,0)$ to make it continuous at the origin.
