Prove that the function f:R2 to R defined by f(x,y) =
{ xy/(x2+y2) if (x,y) not equal to (0,0)
0, if (x,y) = (0,0)
Is not continuous at (0,0)
To verify whether this function is continuous at we need to study the limit of when . When calculating limits of this type (especially in ) , it is very useful to pass into the polar coordinate system, where the limit is replaced by a limit independently of . In polar coordinates :
And we see that this limit depends on and thus the limit does not exist, so the function is not continuous at . For more visibility, we can calculate the limit along which gives and the limit along which gives .
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