   Chapter 2.5, Problem 8E

Chapter
Section
Textbook Problem

Sketch the graph of a function f that is continuous except for the stated discontinuity.Neither left nor right continuous at −2, continuous only from the left at 2

To determine

To sketch: The graph of a function f is continuous except neither left nor right continuous at −2, continuous only from the left at 2.

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.

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:

A function f is discontinuous from the left at a number −2 if limx2f(x)f(2) and a function f is discontinuous from the right at a number −2 if limx2+f(x)f(2). Thus the actual value of f(x) at x=2 is defined in except L(the limit value of f(x) as x approaches −2 or the left and right limit value of f(x) as x approaches −2).

By definition 2, a function f is continuous only from the left at a number 2means that the function f is discontinuous from the right at a number 2. That is, limx2f(x)=f(2) and limx2+f(x)f(2)

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