1549901369-Elements_of_Real_Analysis__Denlinger_

9.3 Implications of Uniform Convergence in Calculus 559

The graph of a typical f n is shown in Fig-

ure 9.11. The pointwise limit of Un} on [O, 1]

is f(x) = x. Each fn is differentiable at 0, and

f~(O) = 0, so f~(O) -+ 0. However, f'(O) = 1.

Therefore, the answer to the second part of

Q#4 is "no." D

, ,

,

We now show how uniform convergence -1

affects the concerns raised in questions

Q#l-Q#4. We address these concerns in the

same order.

, ,

,

, ,

, ,

,

, ,

y

, , ,

,

Figure 9.11

x

Theorem 9.3.5 (Uniform Convergence Permits Interchange of Lim-

its^5 ) :

Suppose Un} converges uniformly to f on a set S - {xo} for some xo ES. If

each f n has a (finite) limit as x-+ Xo then so does f, and we can interchange the

limits. More precisely, if 'Vn EN, lim fn(x) exists, then lim f(x) exists and

x-q x-q

equals n-.oo lim ( lim x--+xo fn(x)). That is, x--+xo lim ( lim n--+oo fn(x)) = n--+oo lim ( lim x---txo fn(x)).

Proof. Suppose f n -+ f uniformly on S - {xo} for some xo E S, and
suppose that each of the functions f n in the sequence has a limit as x-+ xo,

lim fn(x) = Ln ER

X--+Xo

Let r:: > 0. By the uniform Cauchy criterion,

::lno EN 3 m,n 2: no=? llfn -fmll < ~­

Now, 'Vx ES - {xo},

(1)

ILn - Lml ::; ILn - fn(x)I + lfn(x) - fm(x)I + lfm(x) - Lml· (2)

By Equation (1), 'Vk EN, :3<5k > 0 3 0 < Ix -xol < <5k =? lfk(x) - Lkl < ~­
Fix any m, n 2: no and choose any x such that 0 < Ix -xol < min{ Om, On}·
For this x we have

ILn - fn(x)I < ~ and lfm(x) - Lml < ~'as well as lfn(x) - fm(x)I < ~­

Plugging these into (2), we have

ILn - Lml < ~ + ~ + ~ = €.

5. Including one-sided limits.