Suppose f is continuously differentiable on an interval (a, 6). Prove that on any closed subinterval [c, d] the function is uniformly dif- ferentiable in the sense that given any 1/n there exists 1/m (inde- pendent of ro) such that |f(x)-f(xo)-f'(x0)(x-xo)|I < lx– x0|/n whenever r - xo] < 1/m. (Hint: use the mean value theorem and the uniform continuity of f' on [c, d].)
Suppose f is continuously differentiable on an interval (a, 6). Prove that on any closed subinterval [c, d] the function is uniformly dif- ferentiable in the sense that given any 1/n there exists 1/m (inde- pendent of ro) such that |f(x)-f(xo)-f'(x0)(x-xo)|I < lx– x0|/n whenever r - xo] < 1/m. (Hint: use the mean value theorem and the uniform continuity of f' on [c, d].)
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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![**Problem Statement:**
Suppose \( f \) is continuously differentiable on an interval \( (a, b) \). Prove that on any closed subinterval \([c, d]\) the function is uniformly differentiable in the sense that given any \( 1/n \) there exists \( 1/m \) (independent of \( x_0 \)) such that \(|f(x) - f(x_0) - f'(x_0)(x-x_0)| \leq |x-x_0|/n\) whenever \(|x-x_0| < 1/m\).
*(Hint: use the mean value theorem and the uniform continuity of \( f' \) on \([c, d]\).)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe378cba8-62c2-4aa2-b207-bb237650cd78%2Fff9fba9c-dd9d-4245-8a42-26dbb065d6fc%2Fethr5bo_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Suppose \( f \) is continuously differentiable on an interval \( (a, b) \). Prove that on any closed subinterval \([c, d]\) the function is uniformly differentiable in the sense that given any \( 1/n \) there exists \( 1/m \) (independent of \( x_0 \)) such that \(|f(x) - f(x_0) - f'(x_0)(x-x_0)| \leq |x-x_0|/n\) whenever \(|x-x_0| < 1/m\).
*(Hint: use the mean value theorem and the uniform continuity of \( f' \) on \([c, d]\).)*
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