4. Suppose f is defined on R and f satisfies |f(x) − f (y)| ≤ C\x − y|ª for all x, y € R. Here C and a are two given positive constants. Prove that f is uniformly continuous on R.
4. Suppose f is defined on R and f satisfies |f(x) − f (y)| ≤ C\x − y|ª for all x, y € R. Here C and a are two given positive constants. Prove that f is uniformly continuous on R.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Continuity and Uniform Continuity in Real Analysis
#### Problem Statement
4. Suppose \( f \) is defined on \( \mathbb{R} \) and \( f \) satisfies
\[
|f(x) - f(y)| \leq C|x - y|^\alpha
\]
for all \( x, y \in \mathbb{R} \). Here \( C \) and \( \alpha \) are two given positive constants. Prove that \( f \) is uniformly continuous on \( \mathbb{R} \).
#### Explanation
This problem is asking you to prove that a function \( f \), which is defined on the set of real numbers \( \mathbb{R} \) and satisfies a certain condition, is uniformly continuous on \( \mathbb{R} \). The condition provided is an inequality involving the absolute differences between values of \( f \) at different points.
- \( |f(x) - f(y)| \) represents the absolute difference between \( f(x) \) and \( f(y) \).
- \( C \) and \( \alpha \) are given positive constants.
This inequality suggests that the rate of change of \( f(x) \) is controlled by the term \( C|x - y|^\alpha \). Such conditions are often used to ensure that \( f \) does not change too rapidly, a key property needed for uniform continuity.
#### Detailed Solution Outline
1. **Understanding Uniform Continuity**:
- A function \( f \) is uniformly continuous on \( \mathbb{R} \) if, for any \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x, y \in \mathbb{R} \), \( |x - y| < \delta \) implies \( |f(x) - f(y)| < \epsilon \).
2. **Given Condition**:
- We are given that \( |f(x) - f(y)| \leq C|x - y|^\alpha \) for some positive constants \( C \) and \( \alpha \).
3. **Proving Uniform Continuity**:
- To prove that \( f \) is uniformly continuous, we need to find a \( \delta \) for a given \( \epsilon > 0 \).
4. **Steps**:
- Start](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833bf7b1-3e6b-4749-8e88-54090320a3f5%2F936ba3e0-74b6-419e-9964-bc7e06864738%2Funr54r_processed.png&w=3840&q=75)
Transcribed Image Text:### Continuity and Uniform Continuity in Real Analysis
#### Problem Statement
4. Suppose \( f \) is defined on \( \mathbb{R} \) and \( f \) satisfies
\[
|f(x) - f(y)| \leq C|x - y|^\alpha
\]
for all \( x, y \in \mathbb{R} \). Here \( C \) and \( \alpha \) are two given positive constants. Prove that \( f \) is uniformly continuous on \( \mathbb{R} \).
#### Explanation
This problem is asking you to prove that a function \( f \), which is defined on the set of real numbers \( \mathbb{R} \) and satisfies a certain condition, is uniformly continuous on \( \mathbb{R} \). The condition provided is an inequality involving the absolute differences between values of \( f \) at different points.
- \( |f(x) - f(y)| \) represents the absolute difference between \( f(x) \) and \( f(y) \).
- \( C \) and \( \alpha \) are given positive constants.
This inequality suggests that the rate of change of \( f(x) \) is controlled by the term \( C|x - y|^\alpha \). Such conditions are often used to ensure that \( f \) does not change too rapidly, a key property needed for uniform continuity.
#### Detailed Solution Outline
1. **Understanding Uniform Continuity**:
- A function \( f \) is uniformly continuous on \( \mathbb{R} \) if, for any \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x, y \in \mathbb{R} \), \( |x - y| < \delta \) implies \( |f(x) - f(y)| < \epsilon \).
2. **Given Condition**:
- We are given that \( |f(x) - f(y)| \leq C|x - y|^\alpha \) for some positive constants \( C \) and \( \alpha \).
3. **Proving Uniform Continuity**:
- To prove that \( f \) is uniformly continuous, we need to find a \( \delta \) for a given \( \epsilon > 0 \).
4. **Steps**:
- Start
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