Question

Suppose fon (0,1) is uniformly continuous. Show that there is a real number A such that the function F defined by F(O)=A, F(x)=f(x) if x € (0,1), is continuous on (0,1]. (Suggestion: Show first that if {Xn}, Xn € (0,1] has lim xn = 0, then {f(xn)} is a Cauchy no sequence. Then show this sequence has the same limit no matter which {Xn} sequence going to you choose).

Answer 1

And Let dis uniformly continuous on (o]. we prove that lim 8x) exist dinitely Xot let (xu be a sequence en Co, i convergery to o. Then (ar) is a cauchy seen in (ol] Since des unitermly continuens on (ol , for a pre-assigned positive E there enitts a Positive s such that for every pair of mes x, a"in (o satisfying 1x²-x" <g 186x')-(x") KE in R Since and is a cauchy sequence, there exists a natural number k such that lam-an cs & m, nok. it so hlows that for all mn>k, tham Hrem) - Han) KE in R. This shows that & Han] is a canchy sequence in R. am there sure it is convergent let lim Hand = l. no" Let und be another sequence in (a Converging to 0. Then the sequence on-yn) is a sean in last converging to o. Let E70. Since des unisermly cent in (oil 7 a positive & such that for any tovo Pts a,,na E (o] Im-ral <8 184) - 8(x2)) </

Schee {ans Yn) is a sean en (o, i enverging to o there exists a natural number K such that lan-ynleg & nyk. : 1 sean) - Hyn) | <6/2 & nyk. Id(on-l) < / Hyn-Hand + 1 Hom)-l/ <E & nyk. This proves that lim Hyn)=l. Thus for every sequence and is (o, i] cenverging to o, the son { sean)] Amergers to the limit l. This implies lim sess=d. nt to xato How it we define F. [o] → R by FCO - A al. [ay] F(X) = Ha) ir XE (o] clearly then F is contin on [oo : F (o) - lim Hens of