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Q. For what values of p is the function
f(x,y) = { (x+y)^p/(x^2 +y^2) where (x,y) can not equal (0,0)
0 else
continuous on R^2
I'm finding this problem very confusing. Please help. I know that 0 <= (x-y)^2 =
x^2 -2*x*y+y^2 so 2*x*y <= x^2 +y^2
and I have to use the inequality
(x+y)^2 = x^2 +2*x*y +y^2 <= 2*x^2 +2*y^2
How do I proceed from here. Thanks in advance.
Well, you could always use the calculus definition of continuity: the limit of a function must equal its value in any interval in which the function is continuous. You're going to have two issues that I can see: negative values of x+y combined with a fractional exponent p such as 1/2, and making the sure the limit at the origin matches the function value at the origin. These two issues will, I think, restrict the possible values of p. See what that gives you.