MATH 321 Study Guide - Midterm Guide: Uniform Convergence, Medieval Commune, Riemann Integral
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De ne (a) uniform convergence of a sequence of functions (b) an algebra of functions that vanishes nowhere (c) an atlas and a maximal atlas. Give an example of each of the following, together with a brief explanation of your example. If an example does not exist, explain why not. (a) a function f : [0, 1] ir which is riemann integrable on [0, 1] but for which the function. F : [0, 1] ir de ned by f (x) = r x. Pn= cneinx that does not converge in the mean (d) two charts for ( 1, 1) (with the usual metric) that are not compatible. Let , f, g : [a, b] ir with an increasing function. (a) prove that (b) either prove that b (f + g) d b (f + g) d = Z a b f d + b g d . Let f : [0, 1] ir have a continuous derivative.
