Chapter 1: Q. 90 (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 $1$ .
$f (x) = x^{3}$

Short Answer

Expert verified

Ans: $x^{3}$ is continuous in its domain $x \in R$

Step by step solution

Step 1. Given information.

given expression $f (x) = x^{3}$

Step 2. Domain:

Since, $x$ is defined fo all $x \in R$

Step 3. So, check for continuity.

Let $c$ be any real number.

$f$ is continuous at $x = c$

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

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

$L H S = lim_{x \to c} f (x) = lim_{x \to c} x^{3}$

by putting $x = c$

localid="1648044466347" $= c^{3}$

$R H S = f (c) = c^{3}$

Step 4. Since, LHS=RHS

So, Function is continuous at $x = c$

Thus, we can write that

$f (x) = x^{3} c o n t i n u o u s f o r a l l x \in R$

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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 $1$ .
$f (x) = x^{3}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Domain:

Step 3. So, check for continuity.

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)=x3

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Domain:

Step 3. So, check for continuity.

Step 4. Since, LHS=RHS

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Discrete Mathematics

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 $1$ .
$f (x) = x^{3}$