Chapter 1: Q. 91 (page 122)

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$ .
$f (x) = x^{- 2}$

Short Answer

Expert verified

Ans: $f (x) = x^{- 2}$ is continuous on its domain (continuous for all $x \in R - {0}$ )

Step by step solution

Step 1. Given information.

given, $f (x) = x^{- 2}$

Step 2. Domain:

$f (x) = x^{- 2} = \frac{1}{x^{2}}$

At $x = 0,$

$f (0) = \frac{1}{0} = \infty$

Hence, is not defined at $x = 0$

Step 3. So, check for continuity at all points except 0.

Let $c$ be any real number except $0$

assume that $c$ is less than or equal to $1$

$f$ is continuous at $x = c$

if, $lim_{x \to c} f (x) = f (c)$

localid="1648051618930" $L H S = lim_{x \to c} f (x) = lim_{x \to c} \frac{1}{x^{2}} P u t t i n g x = c = \frac{1}{c^{2}}$

localid="1648051960287" $R H S = f (c) = \frac{1}{c^{2}}$

Step 4. Since, LHS=RHS

The function is continuous at $x = c (E x c e p t 0)$

Thus, we can write that

$f$ is continuous for alllocalid="1648052075809" $x \in R - {0}$

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Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$ .
$f (x) = x^{- 2}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Domain:

Step 3. So, check for continuity at all points except 0.

Step 4. Since, LHS=RHS

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Write a delta–epsilon proof that proves that fis continuous on its domain. In each case, you will need to assume that δ is less than or equal to 1.f(x)=x−2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Domain:

Step 3. So, check for continuity at all points except 0.

Step 4. Since, LHS=RHS

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$ .
$f (x) = x^{- 2}$