Category Archives: Continuity

Winter 1995 #3

Problem Statement: Let be continuous, , and exists for every . Let be increasing on . Show is increasing on . Proof: We wish to show that is increasing on which means that for . Since we know that is continuous and … Continue reading

Posted in Analysis, Continuity, Differentiable, Math, MVT | Leave a comment

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 #1

Problem Statement: Let be differentiable on and suppose that exists and is finite. Prove if exists and is finite, then . Proof: Suppose that and that . Since these limits exist we know that these limits will exist, and be the … Continue reading

Posted in Analysis, Cauchy, Continuity, Differentiable, Math, MVT, Sequence | Tagged , , | Leave a comment

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