<s>[INST] Is the function $f$ continuous at $(0,0)$ 
Is the function $f$ continuous at $(0,0)$?
$f(x, y)$ := $\frac{xy}{|x|+|y|}$, if $(x, y)$ $\not= (0, 0)$
and $f(0, 0) := 0$

My attempt:
$f(x, y)$ = $\frac{xy}{x+y}$ if $(x,y) > (0,0)$ and
$f(x, y)$ = $\frac{xy}{-(x+y)}$ if $(x,y) < (0,0)$
So using polar coordinates:
$f(x, y)$ = $\frac{xy}{x+y}$ = $\frac{rcos\theta sin\theta}{cos\theta + sin\theta}$, thus $|f(x,y)| <= r = {(x^2 + y^2)}^{0.5} $ and thus $f$ is continuous.
$f(x, y)$ = $\frac{xy}{-(x+y)}$ = $\frac{-rcos\theta sin\theta}{cos\theta + sin\theta}$, thus $|f(x,y)| <= - r = {-(x^2 + y^2)}^{0.5} $ and thus $f$ is not continuous.
Is my answer correct?
 [/INST] If $x=r\cos\theta$ and $y=r\sin\theta$, then\begin{align}\left\lvert\frac{xy}{\lvert x\rvert+\lvert y\rvert}\right\rvert&=\frac{r^2\left\lvert\cos(\theta)\sin(\theta)\right\rvert}{r\bigl(\lvert\cos\theta\rvert+\lvert\sin\theta\rvert\bigr)}\\&\leqslant\frac r2,\end{align}since $\left\lvert\cos(\theta)\sin(\theta)\right\rvert\leqslant\frac12$ and $\cos\theta\rvert+\lvert\sin\theta\rvert\geqslant1$. So, $f$ is continuous at $(0,0)$.
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