   Chapter 14.4, Problem 45E

Chapter
Section
Textbook Problem

Prove that if f is a function of two variables that is differentiable at (a, b), then f is continuous at (a, b). Hint: Show that lim ( Δ x , Δ y ) → ( 0 , 0 )   f ( a  +  Δ x , b  +  Δ y )  =  f ( a , b )

To determine

To prove: The function f is a function of two variables that is differentiable at (a,b) , then f is continuous at (a,b) .

Explanation

Proof:

Given:

The function is differentiable at (a,b) .

Definition 7:

“If z=f(x,y) , then f is differentiable at (a,b) if Δz can be expressed in the form

Δz=fx(a,b)Δx+fy(a,b)Δy+ε1Δx+ε2Δy , where ε1andε20 .as (Δx,Δy)(0,0) .”

Calculation:

Let the function be z=f(x,y) .

The function f(x,y) is differentiable at (a,b) .

To show the function f(x,y) is continuous at (a,b) , it is enough to prove that,

lim(Δx,Δy)(0,0)f(a+Δx,b+Δy)=f(a,b) .

By using Definition 7,

f(a+Δx,b+Δy)f(a,b)=Δz=fx(a,b)Δx+fy(a,b)Δy+ε1Δx+ε2Δy

f(a+Δx,b+Δy)=f(a,b)+fx(a,b)Δx+fy(a,b)Δy+ε1Δx+ε2Δy

Taking limit on both side as (Δx,Δy)(0,0) ,

f(a+Δx,b+Δy)=f(a,b)+fx(a,b)Δx+fy(a,b)Δy+ε1Δx+ε2Δylim(Δx,Δy)(0,0)f(a+Δx,b+Δy)=lim(Δx,Δy)(0,0)(f(a,b)+fx(a,b)Δx+

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