Answer
$f(x, y)$ is continuous at point (0,0)
Work Step by Step
We have to prove that $f(x, y)$ is continous at (0,0)
\[
\begin{array}{l}
f(x, y)=\frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} \text { for all }(x, y) \neq(0,0) \\
f(x, y)=1 \quad \text { for }(x, y)=(0,0)
\end{array}
\]
So we find the limit of $f(x, y)$ at point (0,0), and if it exists, then the function is continous at (0,0)
\[
\begin{array}{l}
=\lim _{(x, y) \rightarrow(0,0)} f(x, y) \\
=\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}
\end{array}
\]
Let $z=x^{2}+y^{2}$ when $(x, y) \rightarrow(0,0)$; then $z \rightarrow 0$
\[
\begin{aligned}
&=\lim _{z \rightarrow 0} \frac{\sin z}{z} \\
&=1 \\
\because f(0,0) &=\lim _{(x, y) \rightarrow(0,0)} f(x, y)
\end{aligned}
\]
So $f(x, y)$ is continous at (0,0)
