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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
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 \( \mathbb{R} \).
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
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Similar questions
- Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
Numerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY
Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,
