MATH 321 Study Guide - Midterm Guide: Dirichlet Kernel, Equicontinuity, Infimum And Supremum
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Do not open this test until instructed to do so! This exam should have 19 pages, including this cover sheet. No textbooks, calculators, or other aids are allowed. Turn o any cell phones, pagers, etc. that could make noise during the exam. You must remain in this room until you have nished the exam. Use the back of the page if necessary. Let be increasing and f r( ) on [a, b]. Denote by m and m the in mum and supremum of {|f (x)| : x [a, b]} respectively. (a) show that there exists c [m, m ] such that. Z b a f (x)d = c[ (b) (a)]. (b) if, in addition, f is continous on [a, b], prove that there exists x0 [a, b] such that. Z b a f (x)d = f (x0)[ (b) (a)]. Problem 3 (15 points) (a) let be increasing and assume that f r( ) on [a, b].
