Show that $g$ is continuous at $(0,0)$.

Let $g:\mathbb{R^2}\rightarrow \mathbb{R}$ so that, in $M=[0,1]\times[0,1]$,

$$g(x,y)=\begin{cases}\frac{x^2+y^2}{x+y} &\text{ if }x+y \neq 0,\\\\ 0&\text{ if }x+y=0\end{cases}$$

Show that $g$ is continuous at $(0,0)$ in $M$.

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2 Answers

Hint: To find the limit at $(0,0)$ use polar coordinates $x=r\cos(\theta),$ $y=r\sin(\theta)$ and consider taking the limit as $r\to 0$.

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Use polar coordinates. Since $\sin \theta + \cos \theta = \sqrt{2} \sin(\theta + \frac{\pi}{4})$ then for $0 \leq \theta \leq \frac{\pi}{2}$ the denominator is clearly bounded away from zero on $M$.

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