Chapter 10: Q. 93 (page 777)

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

Short Answer

Expert verified

Ans: $f (x) = x^{\frac{1}{2}}$ is continuous on its domain (continuous for all $x \in R$ )

Step by step solution

Step 1. Given information.

Given, $f (x) = x^{1 / 2}$

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 $1$

x = c

if, role="math" localid="1648053742246"

lim_{x \to c} f (x) = f (c)

$L H S = lim_{x \to c} f (x) = lim_{x \to c} \sqrt{x} P u t t i n g x = c = \sqrt{c}$

$R H S = f (c) = \sqrt{c}$

Step 4. Since, LHS=RHS

So, Function is continuous at $x = c$

Thus, we can write that

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

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)=x1/2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Domain:

Step 3. So, check for continuity.

Step 4. Since, LHS=RHS

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Statistics

Applied Mathematics

Discrete Mathematics

Logic and Functions

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