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How to show continuous partial derivatives
f(x,y)= xy/sqrt(x^2+y^2) for (x,y) =/= 0,0
and f(x,y)=0 for (x,y)=0,0
A)Compute partial derivatives with respect to x and y at an arbitrary point (x,y)=/=0,0 and argue that f has continuous partial derivatives on the punctured plane R[sup:mba1081g]2[/sup:mba1081g]{(0,0)}={x,y)?R[sup:mba1081g]2[/sup:mba1081g] I (x,y)=/=(0,0)}
I have found the partials for x and y
fx=y^2x^2/(x^2+y^2)
fy=(x^2y^2)/(x^2+y^2)
I am confused how to argue that the function has partial derivatives. I know how to show the function is continuous (i.e, limit equals the function value) but I don't know how to show continuous partial derivatives.
B) Prove that f is not C[sup:mba1081g]1[/sup:mba1081g] on R[sup:mba1081g]2[/sup:mba1081g] by showing that any one of the partial derivatives with respect to x and y of f(x,y) is not continuous at (0,0)
I am very confused with this.
Do I do any of these problems by substituting mx for y and then y[sup:mba1081g]2[/sup:mba1081g] for x?
Thank you very much.
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