Let K> 0 and let f: RR satisfy the condition f(x)-f(y)| ≤ Kx-y for all x, y € R. Show that f is continuous at every point e E R.
Let K> 0 and let f: RR satisfy the condition f(x)-f(y)| ≤ Kx-y for all x, y € R. Show that f is continuous at every point e E 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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Question
![**Mathematical Analysis: Proving Continuity**
**Problem Statement:**
Let \( K > 0 \) and let \( f : \mathbb{R} \rightarrow \mathbb{R} \) satisfy the condition
\[
|f(x) - f(y)| \leq K|x - y|
\]
for all \( x, y \in \mathbb{R} \). Show that \( f \) is continuous at every point \( c \in \mathbb{R} \).
**Solution Outline:**
To prove continuity at a point \( c \in \mathbb{R} \), we need to show that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \( |x - c| < \delta \), then \( |f(x) - f(c)| < \varepsilon \).
Given the condition:
\[
|f(x) - f(y)| \leq K|x - y|
\]
Applying this to \( f(x) \) and \( f(c) \), we get:
\[
|f(x) - f(c)| \leq K|x - c|
\]
To ensure \( |f(x) - f(c)| < \varepsilon \), set:
\[
K|x - c| < \varepsilon
\]
Thus, we need:
\[
|x - c| < \frac{\varepsilon}{K}
\]
Therefore, we can choose \(\delta = \frac{\varepsilon}{K}\), which satisfies the definition of continuity. Hence, \( f \) is continuous at every point \( c \in \mathbb{R} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57a7d70-87de-4a1f-8104-5b2578062c6c%2F7edad439-9bfa-411e-91d3-b21f2e58addc%2Fdnr8suk_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Analysis: Proving Continuity**
**Problem Statement:**
Let \( K > 0 \) and let \( f : \mathbb{R} \rightarrow \mathbb{R} \) satisfy the condition
\[
|f(x) - f(y)| \leq K|x - y|
\]
for all \( x, y \in \mathbb{R} \). Show that \( f \) is continuous at every point \( c \in \mathbb{R} \).
**Solution Outline:**
To prove continuity at a point \( c \in \mathbb{R} \), we need to show that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \( |x - c| < \delta \), then \( |f(x) - f(c)| < \varepsilon \).
Given the condition:
\[
|f(x) - f(y)| \leq K|x - y|
\]
Applying this to \( f(x) \) and \( f(c) \), we get:
\[
|f(x) - f(c)| \leq K|x - c|
\]
To ensure \( |f(x) - f(c)| < \varepsilon \), set:
\[
K|x - c| < \varepsilon
\]
Thus, we need:
\[
|x - c| < \frac{\varepsilon}{K}
\]
Therefore, we can choose \(\delta = \frac{\varepsilon}{K}\), which satisfies the definition of continuity. Hence, \( f \) is continuous at every point \( c \in \mathbb{R} \).
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