Writing
​Is the following function continous at the origin g(x,y)= siny* (2xy/(x^2+y^2

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Use $\varepsilon -\delta$ definition to check continuity:

Let $\varepsilon$>0 be given.can we find $\delta$>0 such that if

we know that

[img alt="xy

because

[img alt="-1\leq sinx\leq1" src="http://latex.codecogs.com/gif.latex?-1%5Cleq%20sinx%5Cleq1" >

then we can write

$\left | \frac{xysiny}{x^{2}+y^{2}} \right |\leq 1$

we have

$|f(x,y)-0|=\left | \frac{2xysiny}{x^{2}+y^{2}} \right |=2\left | \frac{xysiny}{x^{2}+y^{2}} \right |\leq 2\epsilon$

so

$\delta =2\epsilon$ will work.

therefore it is continous at (0,0)

khakaan (1456)
University of Maryland
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