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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Sketch the graph of a function f that is continuous except for the stated discontinuity.

Removable discontinuity at 3, jump discontinuity at 5

To determine

To sketch: The graph of a function f is continuous except the removable discontinuity at 3 and jump discontinuity at 5.

Explanation

Jump discontinuity: A function f(x) jumps one value to another value at x=a. That is, the limit does not exists at x=a because limxa+f(x)limxaf(x). Also the function f may have any constant value at x=a.

Removable discontinuity: The limit exists at x=a because limxa+f(x)=limxaf(x) and

the actual value of f(x) at x=a is defined. But limxaf(x)f(a).

Calculation:

By the definition of removable discontinuity, the limit exists at x=3 because limx3+f(x)=limx3f(x) and the actual value of f(x) at x=3 is defined. But limx3f(x)f(3).

By the definition of jump discontinuity, the function f(x) jumps one value to another value at x=5. That is, the limit does not exists at x=5 because limx5+f(x)limx5f(x). Also the function f may have any constant value at x=5

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