Chapter 1: Q. 83 (page 122)

Write a delta–epsilon proof that shows that the function $f (x) = |x|$ is continuous. You may find the following inequality useful: For any real numbers $a$ and $b$ , $||a| - |b|| \leq |a - b|$ . (This exercise depends on Section 1.3.)

Short Answer

Expert verified

Hence we proved $\lim_{x \to c} |x| = c$ .

Step by step solution

Step 1. Given Information

We are given the function $f (x) = |x|$ is continuous and we need to write a delta–epsilon proof.

Step 2. Delta–epsilon proof

$f (x) = |x| c = c L = c$

For $ε > 0, δ = ε$ ,

If $0 < |x - c| < δ$ we have,

$||x| - |c|| = |x - c| < δ = ε ∴ \lim_{x \to c} |x| = c$

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Write a delta–epsilon proof that shows that the function $f (x) = |x|$ is continuous. You may find the following inequality useful: For any real numbers $a$ and $b$ , $||a| - |b|| \leq |a - b|$ . (This exercise depends on Section 1.3.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Delta–epsilon proof

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Write a delta–epsilon proof that shows that the function f(x)=xis continuous. You may find the following inequality useful: For any real numbers aand b, a-b≤a-b. (This exercise depends on Section 1.3.)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Delta–epsilon proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Write a delta–epsilon proof that shows that the function $f (x) = |x|$ is continuous. You may find the following inequality useful: For any real numbers $a$ and $b$ , $||a| - |b|| \leq |a - b|$ . (This exercise depends on Section 1.3.)