   Chapter 14.2, Problem 38E

Chapter
Section
Textbook Problem

Determine the set of points at which the function is continuous.38. f ( x , y ) = { x y x 2 + x y + y 2 if  ( x , y ) ≠ ( 0 , 0 ) 0 if  ( x , y ) = ( 0 , 0 )

To determine

The set of points on which the function f(x,y)={xyx2+xy+y2if(x,y)(0,0)0if(x,y)=(0,0) is continuous.

Explanation

Theorem used: Squeeze Theorem

If f , g and h be three functions then g(x)f(x)h(x) and assume that limxag(x)=L=limxah(x) . Then f(x) also approaches to L as x tends to a , limxaf(x)=L

Calculation:

The given function is, f(x,y)={xyx2+xy+y2if(x,y)(0,0)0if(x,y)=(0,0) .

Notice that the given function is a rational function. And the expressions xy and x2+xy+y2 are polynomials.

Since every polynomial function is continuous on 2 the given function is continuous

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