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Chapter 1: Q. 44 (page 120)

Use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f , describe the type of discontinuity and any one-sided discontinuity.

f(x)=x3,ifx≤01−x,if0<x<3x−5,ifx≥3

Short Answer

Expert verified

The function has a jump discontinuity at break point x = 0.

The function is continuous at break point x = 3.

Step by step solution

01

Step 1. Given information.

We have been given a function:

f(x)=x3,ifx≤01−x,if0<x<3x−5,ifx≥3

We have to determine whether this function f is continuous at its break point(s). For each discontinuity of f , we have to describe the type of discontinuity and any one-sided discontinuity.

02

Step 2. Check continuity at break point x = 0.

limx→0− f(x)=limx→0− x3=03=0limx→0+ f(x)=limx→0+ (1−x)=1-0=1limx→0 f(x)=limx→0 x3=03=0

Since both side limits are not equal, the function has jump discontinuity.

This function is left continuous at x = 0.

03

Step 3. Check continuity at break point x = 3.

limx→3− f(x)=limx→3− (1−x)=1−3=-2limx→3+ f(x)=limx→3+ (x−5)=3−5=-2limx→3 f′(x)=limx→3 (x−5)=3−5=-2

Since all the values are equal, the function is continuous at x = 3.

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