   Chapter 2.5, Problem 6E

Chapter
Section
Textbook Problem

Sketch the graph of a function f that is continuous except for the stated discontinuity.Discontinuities at –1and 4, but continuous from the left at –1 and from the right at 4

To determine

To sketch: The graph of a function f is continuous except at x=1 and x=4, but continuous from the left at x=1 and from the right at x=4.

Explanation

Definition used:

1 A function f is continuous from the right at a number a if limxa+f(x)=f(a).

2. A function f is continuous from the left at a number a if limxaf(x)=f(a).

Note 1: The function f is said to be discontinuous at x=a if anyone of the following conditions does not satisfied.

• f(a) is defined
• The limit of the function at the number a exists.
• limxaf(x)=f(a)

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.

Calculation:

By definition 1, a function f is continuous from the right at a number 4if limx4+f(x)=f(4) and by note 1, a function f is continuous everywhere except x=4 means that limx4f(x)f(4). This implies that, the left hand and right hand limits are not equal. That is, limx4f(x)limx4+f(x) and the function f is discontinuous from the left at a number 4.

By definition 2, a function f is continuous from the left at a number −1 if limx1f(x)=f(1) and by note 1, a function f is continuous everywhere except x=1 means that limx1f(x)f(1). This implies that, the left hand and right hand limits are not equal. That is, limx1f(x)limx1+f(x) and the function f is discontinuous from the right at a number −1

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