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2 (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. S _ ɁAT V c J o A f j ɗl X Ȉ Ă ܂ B l C x g A ̔ p ̃f U C T V c ܂ŁA I W i ȃf U C g ɂ Ċy ŗ~ Ɠ X l Ă ܂ B. F i1 g ̂āj J R y u b N zARTIRAL t u Moist A e B UV C X g 10 SHOBI ʔ̂Ŕ Ȃ烁 K l X p O v ʔ̃T C g ߁B5,400 ~ ȏ w ő I R r j 㕥 A X E R r j A R ^ N g ( w ) A z A K l ȂǃT r X I.

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