BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2, Problem 15RQ
To determine

To find: Whether the statement “If f is continuous at a , then f is differential at a ” is true or false.

Expert Solution

Answer to Problem 15RQ

The statement “If f is continuous at a , then f is differential at a ” false.

Explanation of Solution

Given information:

The given statement is “ f is continuous at a ”.

Calculation:

Let us assume a function f(x)=|x| .

Check the continuity of function for x<0 .

  ltx0f(x)=ltx0|x|ltx0f(x)=ltx0(x=)ltx0f(x)=lth0(0h)lth0=0

Check the continuity of function for x>0 .

  ltx0+f(x)=ltx0+|x|ltx0+f(x)=ltx0+(x)ltx0+f(x)=lth0(0+h)lth0+=0

Here, left hand limit is same as right hand limit and f(0)=0 .

So, the function is continuous at 0 .

Evaluate left hand derivative of the function.

  Lf(0)=limx0f(x)f(0)x0=limx0|x|0x0=limx0xx=1

Evaluate right hand derivative of the function.

  Rf(0)=limx0+f(x)f(0)x0=limx0+|x|0x0=limx0+xx=1

Since left hand derivative and right hand derivative is not same the given function is not differentiable.

Therefore, the statement “If f is continuous at a , then f is differential at a ” is false.

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