1549901369-Elements_of_Real_Analysis__Denlinger_

9 .1 Families of Tunctions and Pointwise Convergence 545

Examples 9.1. 7 We give here some examples of sequences {fn} and their

pointwise limits. Exercise 2 asks you to prove the claims made here.

(a) Let S = [O, 1] and fn(x) = xn. The Y

graphs of fn for n = 1, 2, 3, 4 are shown in

Figure 9.1. It is clear from Theorem 2.3.7

that the pointwise limit of the sequence

{! n} is the function

f ( x) = { 0 ~f 0 ::; x < 1;

l1fx=l.

(See Figure 9.1.) Thus we say

fn --t f (pointwise) on [O, l].

(b) Let S = [O, 1] and define fn : [O, 1] --t JR

b f (x) = { 2n - 2n
2
x if 0::; x::; ~;
Y n 0 if .1 n < X < - 1.

The graph of a typical f n is shown in Figure 9.2.
It is clear that the pointwise limit of {f n} on
(0, 1] is f(x) = 0, but the pointwise limit does
not exist on [O, 1] because lim fn(O) does
n->oo
not exist.

(c) Let S = [-1, 1] and define fn:

[-1, 1] --) JR

{

.!. if [x[ < .!..
by fn(x) = [~[ if ~ <-[; ['::; 1.

The graph of a typical f n is shown in
Figure 9.3.
It is clear that the pointwise limit of

x

Figure 9. 1

y

(0, 2n)

1. II x

Figure 9.2

y

1. II

{fn} on [- 1, 1] is f(x) =[xi- --+-l---_-+l.--+--+1.----+--x

IZ ll

Figure 9.3

(d) Let S = [O, 1] and define fn : [O, 1] --t JR

{

1 if x = 0, or x = T for some

by fn(x) = relatively prime m, k EN,

where k ::; n;

0 otherwise.