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

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help me dear

Jul 3rd, 2014

Use \varepsilon -\delta definition to check continuity:

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

 0<p><strong>  then</strong></p><p><strong> [img alt=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

 xysiny<p><strong>that is</strong></p><p><strong> [img alt=

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)


Jul 3rd, 2014

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