midterm2 - { f n : n N } is equicontin-uous. 5. In each of...

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Modern Analysis 2 Midterm Answer at most four questions. 1. Let f be strictly increasing and continuous on [0 ,a ] and differentable on (0 ,a ) with f (0) = 0. (i) Prove that if g ( u ) := Z u 0 f ( t )d t + Z f ( u ) 0 f - 1 ( t )d t - uf ( u ) then g ( u ) = 0 for all u [0 ,a ]. (ii) Deduce that if 0 < b 6 f ( a ) then ab 6 Z a 0 f + Z b 0 f - 1 . 2. Let f n ( t ) = t (1 - t 2 ) n for n N and 0 6 t 6 1. (i) Calculate R 1 0 f n ( t )d t . (ii) Does the sequence ( f n ) n N converge pointwise on [0 , 1]? Uniformly? 3. Let f : R R be differentiable, with uniformly continuous derivative. For n > 0 and t R define g n ( t ) = n h f ( t + 1 n ) - f ( t ) i . Prove that g n f 0 uniformly on R . 4. (i) State the (abstract) Arzela-Ascoli theorem. (ii) Let ( f n ) n N be a uniformly convergent sequence of continuous real-valued functions on a compact metric space; prove that
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Unformatted text preview: { f n : n N } is equicontin-uous. 5. In each of the following cases, decide for which real a &lt; b the indicated family of functions is (uniformly) dense in C ([ a,b ]). (i) All polynomials of the form c + c 1 t 2 + + c n t 2 n . (ii) All functions of the form c + c 1 t 1 / 5 + + c n t n/ 5 . (iii) All functions of the form c + c 1 e t + + c n e nt . Here, n runs over all nonnegative integers and the coecients c ,c 1 ,...,c n are real. 1...
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