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Assignment 5 Multivariate Calculus Math 251 (Fall 2010) A1. For what values of p is ths function f ( x, y ) = braceleftBigg ( x + y ) p x 2 + y 2 ( x, y ) negationslash = (0 , 0) 0 otherwise continuous on all of R 2 ? Hints: For what values of p is g ( x ) = x p continuous on R ? Try to bound the denominator by a term that looks like radius. Solution This problem is particularly challenging to solve completely. We claim that f ( x, y ) is continuous on R 2 if and only if p > 2. We only need to check the continuity of f at (0 , 0) since elsewhere, it is a rational function. We are given that f (0 , 0) = 0 so we need to see whether or not lim ( x,y ) (0 , 0) f ( x, y ) = 0. That is, we need to check whether ( x + y ) p x 2 + y 2 goes to 0 as radicalbig ( x 0) 2 + ( y 0) 2 goes to zero. First we show that if p > 2, then f is continuous at (0 , 0). We will need to show that the value of ( x + y ) 2 can not be much greater than the value of x 2 + y 2 , so we calculate as follows. Since 0 ( x
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