[Math] Is function continuous $x\sin(y)/(x^2+y^2)$

calculuscontinuityfunctions

I have the following function and I can't seem to prove that it is not continuous:
$
f(x,y) =
\begin{cases}
0, & {(x,y) = (0,0)} \\
x\sin(y)/(x^2+y^2), & \text{else} \\
\end{cases}$

Best Answer

Using the Taylor series: $$\sin x\sim_0x$$ we find $$f(x,x)\sim_0\frac12\quad;\quad f(x,2x)\sim_01$$ so $f$ isn't continuous at $(0,0)$. Obviously $f$ is continuous on $\Bbb R^2\setminus\{(0,0)\}$.

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