Proposition 4.1.2. If ƒ : A → Rn satisfies the lower-Lipschitz condition on E, i.e., if there exists of some c> 0 such that c|x − y| ≤ f(x) = f(y) for all x, y ≤ E, then f is 1-1 on E, and hence its inverse function g = f-1 exists on f(E). Moreover, g: f(E) → R is Lipschitz on f(E).
Proposition 4.1.2. If ƒ : A → Rn satisfies the lower-Lipschitz condition on E, i.e., if there exists of some c> 0 such that c|x − y| ≤ f(x) = f(y) for all x, y ≤ E, then f is 1-1 on E, and hence its inverse function g = f-1 exists on f(E). Moreover, g: f(E) → R is Lipschitz on f(E).
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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Need help to prove the following.
![**Proposition 4.1.2**: If \( f : A \rightarrow \mathbb{R}^n \) satisfies the lower-Lipschitz condition on \( E \), i.e., if there exists some \( c > 0 \) such that
\[ c|x - y| \leq |f(x) - f(y)| \quad \text{for all } x, y \in E, \]
then \( f \) is 1-1 on \( E \), and hence its inverse function \( g = f^{-1} \) exists on \( f(E) \). Moreover, \( g : f(E) \rightarrow \mathbb{R}^n \) is Lipschitz on \( f(E) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5870f95c-14fe-43b9-9aa2-c4d29aad7e05%2F38a121bf-4dcc-43a1-8a0b-f7c45fcb0920%2Fxz47857_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Proposition 4.1.2**: If \( f : A \rightarrow \mathbb{R}^n \) satisfies the lower-Lipschitz condition on \( E \), i.e., if there exists some \( c > 0 \) such that
\[ c|x - y| \leq |f(x) - f(y)| \quad \text{for all } x, y \in E, \]
then \( f \) is 1-1 on \( E \), and hence its inverse function \( g = f^{-1} \) exists on \( f(E) \). Moreover, \( g : f(E) \rightarrow \mathbb{R}^n \) is Lipschitz on \( f(E) \).
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