Chapter 1: Q. 89 (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 in its domain $x \in R$ .

Step by step solution

Step 1. Given information.

given,

$f (x) = x^{2}$

Step 2. Domain

Since $x$ is defined for 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, localid="1648043730196" $lim_{x \to c} f (x) = f (c)$

localid="1648043796713" $L H S = lim_{x \to c} f (x) = lim_{x \to c} x^{2}$

by putting $x = c$

localid="1648044660051" $= c^{2}$

localid="1648044676259" $R H S = f (c) = c^{2}$

Step 4. Since, LHS =RHS

So, Function is continuous at $x = c$

Thus, we can write that

$f (x) = x^{2}$ is continuous for all $x \in R$

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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 $1$ .
$f (x) = x^{2}$

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)=x2

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

Logic and Functions

Pure Maths

Geometry

Calculus

Discrete Mathematics

Probability and Statistics

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 $1$ .
$f (x) = x^{2}$