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Uð {e € "³$‹ & º(•ê*›«,¡i¦ 0«©2°×4µ÷6»#8ÀBÅ`Ë*>Ð^@Õ~BÚ DßÊFäéHê JïÜLõ Nú(Pÿ@R pT 'V ±X ÞZ =\ m^ #Y` (Èb d 3Îf 8ñh > j CBl Hfn Mˆp R¬r WÑt \óv byx h z m( rK~ wk€ ‚ „ ‡L† Œµˆ 'ÓŠ –ýŒ œ'Ž ¡f ¦ ' «°" °ËV Ww O w U t AP AirP assengers plotAP ylab P assengers s O v w D t x m OQO O Hw windo w s R u R QO Qo D C U q y m Q H p Y L pm W Time Passengers (1000's) 1950 1952 1954 1956 1958 19602 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all x
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Fz kgand fx ngSince fz kgconverges, so does fa mg, contradicting the assumption that fx nghas no convergent subsequence (Note the similarities with the solution of Exercise 7 from Homework 3) 394 Let Mbe a metric space such that Mis a nite setN E B ̏ W Ȃ I I R X v f B X f B Y j R X ` s X J ` V { 3 _ Z b g I t V _ W b p t l n E B ` b v ƃf N X WaltDisney ߑ ϑ p L g C x g p e B O b Y e } p N CHIP'N'DALE Ȃ肫 I X X xPressure_Reducing_ValveZurn_WilkinsModel_500XL_Threaded Zurn Wilkins



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