Chapter 1: Q. 92 (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 $1$ .
$f (x) = x^{- 1 / 2}$

Short Answer

Expert verified

Ans: $f (x) = x^{- \frac{1}{2}}$ is continuous on its domain (continuous for all $x \in R - {0}$ .

Step by step solution

Step 1. Given information.

given,

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

Step 2. Domain:

$f (x) = x^{- \frac{1}{2}} = \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{0}} = \infty$

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="1648053903983">LHS=limx→c f(x)=limx→c 1xPuttingx=c=1c

localid="1648054533319" $R H S = f (c) = \frac{1}{\sqrt{c}}$

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 all $x \in R - {0}$

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

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−1/2

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

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Theoretical and Mathematical Physics

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Geometry

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