Monthly Archives: June 2011

Winter 2008 #5

Problem Statement: Let . Prove is Riemann Integrable on and .  Proof: First note that each is continuous on and since is a sum of continuous functions it follows that is continuous on . Furthermore, since is a compact set it … Continue reading

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April 2009 #2

Problem Statement: Suppose is a real-valued differentiable function such that for every . Show there is at most one such that . Proof: Suppose to the contrary that there are two such values such that and . Wlog assume that . Since … Continue reading

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Fall 2004 #4

Problem Statement: Let be a continuous function on such that is differentiable at all points of except possibly at a single point . If exists show that exists and that . Proof: by definition. Consider: . Applying the definition stated above we … Continue reading

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Fall 2004 #1

Problem Statement: (a) Given a set define what it means for a point to be an interior point of . (b) For let . Show that is open in . (a)Definition: A point is an interior point of if there exists … Continue reading

Posted in Analysis, Topology | Leave a comment

Spring 2008 #5

Problem Statement: Let be a sequence of continuous functions defined on and a real valued function defined on . (a) Prove if uniformly then and (b) Does this hold is pointwise? Prove or give a counter example. (a) Proof: We are given … Continue reading

Posted in Analysis, Continuity, Math, Uniform Convergence | 1 Comment