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.