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Chapter 1: Q. 47 (page 121)

Use graphs to determine if each function is continuous at the given point x = c.

f(x)=2−x,    ifxrationalx2,    ifxirrational,c=1

Short Answer

Expert verified

The function is continuous at x = 1.

Step by step solution

01

Step 1. Given information.

We have been given a function as:

f(x)=2−x,    ifxrationalx2,    ifxirrational,c=1

We have to determine if this function is continuous at the given point x = c.

02

Step 2. Draw the graph.

03

Step 3. Analyze the graph for the continuity of the function at x = c.

From the graph, we can see that:

For the rational values of x,limx→1 f(x)=1

For the irrational values of x, limx→1 f(x)=1

These two values are the same.

Therefore, this function is continuous at x = 1.

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Most popular questions from this chapter

Explain why the Intermediate Value Theorem allows us to say that a function can change sign only at discontinuities and zeroes.

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

limx→-5(3x-2).

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

limx→0-f(x)=-1,limx→0+f(x)=1,f(0)=0.

Write each of the inequalities in interval notation:

0<|x-2|<0.1

For each functionf graphed in Exercises23–26, describe the intervals on whichf is continuous. For each discontinuity off, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

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