Math 413, Fall semester, 2007Bigtimes Homework set 6, handed out September 27, due October 4Remember that there is a prelim scheduled for October 4, 7:30 pm. I don’t know theroom yet, but I will announce it as soon as I do. The curriculum is the text throughChapter 4.You should have read everything through Chapter 4.Here are some review exercises.Problem 1.Consider the functionf:R→Rdeﬁned byf(x)=‰1qifx∈Qandx=pqwithp, qcoprime0ifxis irrational.Show thatfis continuous at all irrationals.Problem 2.A functionf:[a, b]isconvexif for allx, ysatisfylinga≤x<y≤band allt∈(0,1) we havef(tx+(1-t)y)≤tf(x)+(1-t)f(y).(a) Show that every convex function is continuous.(b) Show that iffis convex, then so isef.Show thatfa, b]→Ris convex if and only if it satisﬁes
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