1549901369-Elements_of_Real_Analysis__Denlinger_

9.3 Implications of Uniform Convergence in Calculus 563

Then,

lfn(x) - fm(x)I :=::; lfn(xo) - fm(xo)I +Ix -xol lf~(t) -f,',,(t)I

E E

< 2 + (b - a) 2(b - a) = c:.

Thus, m,n;::: max{n1, n2} => llfn -fmll < c:. Therefore, Un} converges uni-

formly on [a , b]. Define

f = lim fn· (5)

n-+oo

Part 2: Remarks on Part l.

(1) Xo may be any point of [a, b] for which {fn(xo)} converges. Since fn---+ f

uniformly on [a, b], Equation ( 4) holds for all x 0 E [a, b].

(2) Each f n is differentiable on [a, b], so it is continuous there. Thus, by

Corollary 9.3.6 , f is continuous on [a, b].

Part 3: Let xo E [a, b]. Define functions 9n and g on [a, b] by

{

fn(x) - fn(xo) if x I-Xo}

9n(x) = x -Xo and

f~(xo) if x = xo

{

f(x) - f(xo) if x I-xo}

g(x)= x-xo.

f'(xo) if x = xo

Since each f is differentiable at xo,
lim 9n(x) = f~(xo) = 9n(xo).
x---+xo
Thus, each 9n is continuous at Xo. Also, Vx E [a, b],

{

[fn(x) - fn(xo)] - [fm(x) - fm(xo)]

x - xo

9n(x) -9m(x) =

f~(xo) - f'(xo)

= { [fn(x) - fm(x)~ -=-[~:(xo) -fm(xo)]

f~(xo) - f'(xo)

if X f. Xo}

if X = Xo

if X -=fa. Xo}.

if X = Xo

(6)

Let c: > 0. Since {!~} converges uniformly on [a, b], :3 n3 E N 3 m, n 2:
n3 => II!~ -f.'nll < c:.