Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Suppose that f: R→ R is a function. Suppose that there exists a constant c< 1 so that for all x, y E R we have f(x) = f(y)| < cx - y. Prove that f is continuous on R.

Suppose that f: R→ R is a function. Suppose that there exists a constant c< 1 so that for all x, y E R we have f(x) = f(y)| < cx - y. Prove that f is continuous on R.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%
**Problem Statement:** Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is a function. Suppose that there exists a constant \( c < 1 \) so that for all \( x, y \in \mathbb{R} \) we have \( |f(x) - f(y)| < c|x - y| \). Prove that \( f \) is continuous on \( \mathbb{R} \).
Transcribed Image Text:**Problem Statement:** Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is a function. Suppose that there exists a constant \( c < 1 \) so that for all \( x, y \in \mathbb{R} \) we have \( |f(x) - f(y)| < c|x - y| \). Prove that \( f \) is continuous on \( \mathbb{R} \).
**Theorem 4.3.2 (Characterizations of Continuity)** Let \( f : A \to \mathbb{R} \), and let \( c \in A \). The function \( f \) is continuous at \( c \) if and only if any one of the following three conditions is met: (i) For all \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that \( |x - c| < \delta \) (and \( x \in A \)) implies \( |f(x) - f(c)| < \epsilon \); (ii) For all \( V_{\epsilon}(f(c)) \), there exists a \( V_{\delta}(c) \) with the property that \( x \in V_{\delta}(c) \) (and \( x \in A \)) implies \( f(x) \in V_{\epsilon}(f(c)) \); (iii) For all \( (x_n) \to c \) (with \( x_n \in A \)), it follows that \( f(x_n) \to f(c) \). If \( c \) is a limit point of \( A \), then the above conditions are equivalent to (iv) \(\lim_{x \to c} f(x) = f(c)\).
Transcribed Image Text:**Theorem 4.3.2 (Characterizations of Continuity)** Let \( f : A \to \mathbb{R} \), and let \( c \in A \). The function \( f \) is continuous at \( c \) if and only if any one of the following three conditions is met: (i) For all \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that \( |x - c| < \delta \) (and \( x \in A \)) implies \( |f(x) - f(c)| < \epsilon \); (ii) For all \( V_{\epsilon}(f(c)) \), there exists a \( V_{\delta}(c) \) with the property that \( x \in V_{\delta}(c) \) (and \( x \in A \)) implies \( f(x) \in V_{\epsilon}(f(c)) \); (iii) For all \( (x_n) \to c \) (with \( x_n \in A \)), it follows that \( f(x_n) \to f(c) \). If \( c \) is a limit point of \( A \), then the above conditions are equivalent to (iv) \(\lim_{x \to c} f(x) = f(c)\).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS