Category Archives: Riemann Integrable

Suz and Mike had a question

Problem Statement: If is bounded with finitely many discontinuities on then is Riemann Integrable on . Proof: Let be the number of discontinuities of on and let be such that for every . We know such an exists since is bounded. Since … Continue reading

Posted in Analysis, Math, Riemann Integrable | 2 Comments

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

Posted in Analysis, Compact, Math, Riemann Integrable, Series of Functions, Uniform Convergence | Leave a comment

More practice with proving a function is Riemann Integrable

If and if is Riemann Integrable on both and , then is Riemann Integrable on . Proof: Since we know is Riemann Integrable on and it follows that is bounded on those intervals as well. Let be the maximum of those … Continue reading

Posted in Analysis, Math, Riemann Integrable | 4 Comments

Spring 2003 #1

Problem Statement: (a) State a definition of a Riemann Integral. (b) Use this definition to prove if are both bounded, Riemann Integrable, and defined on , then is Riemann Integrable on . (a) A bounded function is Riemann Integrable on … Continue reading

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Winter 2004#5

Problem Statement: Let be defined on . (a) Find . (b) State a definition of Riemann Integrable and use it to prove or disprove is Riemann Integrable on . (a) Note that for and that for . We know that if … Continue reading

Posted in Analysis, Math, Riemann Integrable, Sequence of Functions | 2 Comments

September 1991 #2

Problem Statement: Suppose that is a continuous real valued function on such that for every and . Show . Proof: Since is continuous on we know is Riemann Integrable. We are given that and since is continuous we know that for every … Continue reading

Posted in Analysis, Continuity, Riemann Integrable | 3 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