Category Archives: Continuity

UIC Master’s Exam- Fall 2007 R2

Problem Statement:  is continuous on . Given that (a) prove has continuous derivative on (Hint: start with the change of variable u=x+t). (b) Given and show that there exists a such that for every . Solutions: (a)Proof: First let us start … 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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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

Spring 2004 #5

Problem Statement: Let be differentiable. Let such that and . If is between and then there exists such that . Proof: Wlog assume . Fix . Define . Then . Note that and since . and so for sufficiently close to , … Continue reading

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Spring 2008 #2

Problem Statement: Let be a twice differentiable real valued function defined on . Suppose that with and . Prove there exists a such that . Proof: Since is twice differentiable on we know that is differentiable on , which implies that … Continue reading

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Spring 2008 #1

Problem Statement: Let be a compact subset of and a continuous function. Show there exists a such that for every . Proof: Since is compact and is continuous it follows that is compact. Bolzano-Weierstrass gives us that is closed and bounded. This means … Continue reading

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Winter 1993

Problem Statement: Suppose that converges to uniformly and converges to uniformly on where each is continuous on . Prove that converges uniformly to on . Proof: Since is a compact set and are each continuous on it follows that each is bounded … Continue reading

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