Category Archives: Uniform Continuity

May 1990 #4

Problem Statement: Using properties of the Riemann Integral show that if is Riemann Integrable on and then is uniformly continuous on . Proof: We wish to show that for every there exists a such that for every in it follows that . … Continue reading

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Winter 1995 #4

Problem Statement: Let be uniformly continuous and let for every . Show converges uniformly. Proof: First I claim that converges pointwise to . Let . Then . Since is continuous and converges to it follows that . This holds for all … Continue reading

Posted in Analysis, Sequence, Sequence of Functions, Uniform Continuity, Uniform Convergence | Leave a comment

Fall 1991 #5

This begins my study of sequences and series of functions. These bad boys have a tendency of tripping me up so I’ve decided to spend some quality time learning their ways. Problem Statement: Let and let be a sequence of real-valued functions on … Continue reading

Posted in Analysis, Math, Sequence of Functions, Uniform Continuity | 2 Comments

September 2001 #4

Problem Statement: Assume is a bounded Riemann Integrable function on . Set . Prove that is uniformly continuous on . Proof: Let such that for every . Let and let . Then for it follows that . We are given that is … Continue reading

Posted in Analysis, Riemann Integrable, Uniform Continuity | Tagged , , | 2 Comments

Winter 2004 #4

Problem Statement: Suppose that is differentiable on the bounded interval . Suppose further that there is a constant such that for every . Show that exists and is finite. Note: When I read this question the first time I thought the … Continue reading

Posted in Analysis, Cauchy, Continuity, Differentiable, Math, MVT, Sequence, Uniform Continuity | Tagged , , | 4 Comments