Advanced Calculus fi..

Chapter 6 Infinite Series 41 9Figure 6.10The sequence f,,(x) = x".where Sn(x) = ul(x) + - - . + u,(x); this is possible since the series is uniformlyconvergent. The function SN(x), as sum of ajnite number of continuous functions,is itself continuous. One can hence choose a 6 such thatBy (6.28), one hasIS&)1- SN(XO)I c j .~for IX - X OI < 8. (6.29)HenceIf (x) - f (xo)l = I f (x) - SN(X)+ SN(X) - SN(XO) + SN(XO) - f (xo)~11 1 1< - E + -E + -6 = E , for JX - X OI < 6,3 3 3by (6.29) and (6.30). Thus continuity is proved. ' \Remark 1. The property of convergence alone, for a series of continuous functions,does not guarantee continuity of the sum. This is seen by the example:Here the nth partial sums form the sequence Sn(x) = xn plotted in Fig. 6.10. As waspointed out in Section 6.12, this sequence does not converge uniformly. The sum ofthe series is 0 for 0 5 x < 1 and 1 for x = 1; there is a jump discontinuity at x = 1.Remark 2. Theorem 31can be interpreted in terms of sequences as follows: If Sn(x)is a sequence of functions all continuous for a ( x 5 b and this sequence convergesuniformly to f (x) for a 5 x 5 b, then f (x) is continuous for a I x 5 b; furthermore,if a 5 xo 5 b, thenlirn s.(x)] = lirn [ lim $(x)].n+m x+xo1 (6.31)