Question

2.)

(b): Prove or disprove the following problems. 1. Suppose fn(x) is uniformly convergent to fon D= [a, b]. Let ce [a, b]. Is fr uniformly convergent to f on D1 = (a, and/or D2 = (c, b)? = (a, and D2 = [c, b). Is in 2. Let a <c<b. Suppose fn(x) is uniformly convergent to f on D uniformly convergent to f on D = (a,b). 3. Suppose that fn(a) is uniformly convergent to fon , i=1,2,... Is fn uniformly convergent to on 1 with finite m. How about the uniform convergence of 'n on

Answer 1

Definition of uniform convergered! Let (in) hen be a sequence - of function defireed on [a, b]. Then In Uniformsly convergent to f on [an if for aro The IN 8 - Ifma) -- fWI<a , f man of x@[a,b], [1] Yes in uniformly convenent of on D = [ae] and D = [0,b] from the definition it is clear that for a>0 then - Imbs-flegfeo , & man , & xc [a,b] and [a,b] = [a, c] u [e, b} . => man- f(x, caut mon if Xe [ac] and also lfmas - faglsa & mom ara ele, b] [2] but fna's uniformly convergent teef on P,= [ac] and D₂ = [a, b] . Then & uxo 7 h el sito mexi-fear I co & mana and & xe [ac] Alsce, of Goxo 7 hein sito - (A) - Ifm (x) - foto & mo, ha and roll, b7 het M = maxçh,, na} - BT Then from Alan (B) it is clear that & >0 FNCN git. fm (xg- falco tms, N and f xc [a,b]

m 18] Similar to part (2) given that fn uniformly convergen't to f on Ij, i = 1, 2, 3 ... Then & axo 7 ni i t= 1, 2, 3: sit. If mea ) -f2037 <co # mahi and H XE Ii If we take max {n isism} = N then # Como INE IN siti fincas - f(x) .cw . * m> N and, & XE U Ii for the case Uli it is not trace take a sequenced fuka)= ah c ond het Ij = 0, 1- Then fu(x) = xh convergent tafzo uniformely on each IN But UT, = [0, 17 .. [=1 . . C1 X=1 Then it is not uniformly convergent te f(x) = } Barn foxe[0,1) it is not a continuous function Which is a contradiction tee the theorom. Because tharam says that if he is a sequence of continuous function defined on a compact sut [aub] Convergent uniformly te for then f is edsce a continuous function ALLIERS