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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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