Math 310 -hw8 solutions18.4, 18.6, 18.10;Friday, 6. Nov 200918.4 LetS⊂Rand suppose there is a sequence (xn)ninSthat converges to a numberx0/∈S. Show that there existsan unbounded continuous function onS.We definef:S→Rbyf(x) :=1x-x0forx∈S.Thenfis continuous, because it is the composition of the two continuous functions:x7→x-x0andy7→1/y.We now verify thatfis unbounded. LetM >0 be given. Then sincexn→x0there isNsuch that for alln∈N,n > Nimplies that0<|xn-x0|<1M.Letn∈Nand suppose thatn > N. Then since|f(xn)|=1|xn-x0|>11/M=M,fis unbounded onS.18.6 Prove thatx= cosxfor somexin (0,π2).We suppose that the functionx7→cosxis continuous onR. Letfbe theR-valued function defined byf(x) =x-cosxforx∈R. Thenfis continuous since it is the difference of continuous functions. Observe thatf(0) =-1<0<π2=fπ2.Sincefis continuous on [0,π2] the Intermediate Value Theorem (18.2) applies and we conclude that there isx∈(0,π2) such thatx-cosx=f(x) = 0. Hence,x= cosxas desired.18.10 Suppose thatfis continuous on [0,2] and thatf(0) =f(2). Prove that there existx, yin [0,2] such that|y-x|= 1and
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