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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

If f is continuous at a, then f is differentiable at a.

To determine

Whether the statement, “if f is continuous at a then f is differentiable at a” is true or false.

Explanation

Theorem used: If f is continuous at b and limxag(x)=b, then limxaf(g(x))=f(b).

Theorem used: If f is differentiable at a then limxa+f(x)=L=limxaf(x).

Definition of limit: The limit of the function is defined as limh0f(x+h)f(x)h.

Result used: The limit of the function limx0|x|x=1.

The function f is continuous at a, but not differentiable at a.

Because, the right and left derivatives of a function are not equal.

Example:

Let the function f(x)=|x| is continuous at x=0, but not differentiable at x=0.

Calculation:

The limit of the function is defined as limh0f(x+h)f(x)h

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