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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.

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It follows that f c x f t x ash Since f c x g x andg is integrable we obtain by It follows that f c x f t x ash since f c x g x andg School Syracuse University;. A N Z X } b v B F s { ́B C X g ^ { f B P A z y W B { i J C v N e B b N ƃ{ f B P A E t P A E G X e E p h i W ̂ X ł B @ F s { s h 318 @ 啟 r PF B waisu@quartzocnnejp. 7ÿsß› “ ­ÿ~ýúu ç½»ž{îÙïËû F¼q`˜–“›G “§È0l =­¤xV¼ýåõXŸ ×;mVEö¢_ÿÐ àÈÁþ½Å³RÇžûêþ¬ŸÅzMý’ÚV•F½ `x9ÖÿZ¿b¹=I™ô2€s â05µX’¡±j âq¾âÀ‚ÚŽV0‚ ù Ž7,X¼ª©¯ø ¿ ¥ °íin¬m.

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