1549901369-Elements_of_Real_Analysis__Denlinger_

546 Chapter 9 11 Sequences and Series of Functions

The graph of a typical f n is shown in Fig-
ure 9.4. It is clear that the pointwise limit of
{fn} on [O, l] is the Dirichlet function
f ( x) = { 1 i.f x i.s i:atio~al;
0 if x is irrational. 0

The examples we have just seen, along
with additional examples provided in the ex-
ercises below, suggest that functions that are
pointwise limits of sequences of functions do
not necessarily share important features of the
functions in the sequence. In Example 9.l.7(a)

y

1 ...............

Figure 9.4

x

each function fn is continuous on [O, l] but the pointwise limit function f is
not continuous at l. In Example 9.l.7(c) each function fn is differentiable at 0
but the pointwise limit function f is not. In Example 9.l.7(d) each function fn
is integrable on [O, 1] but the pointwise limit function f is not. (See Example
7.2.10.)
In Exercises 6 and 7 below, we shall see sequences {! n} of integrable func-
tions that converge pointwise to an integrable function f for which lim t f n '1-
n~cx::> a
t a n-.oo lim f n· In Exercise 8 we shall see a sequence Un} of differentiable functions

that converges pointwise to a differentiable function f for which f~ f+ f'.
Thus, while pointwise convergence is essential in establishing the limit func-
tion, it alone does not guarantee that the limit function inherits "nice" prop-
erties such as continuity, differentiability, and integrability from the functions
in the sequence. In Section 9.2 we shall discuss a stronger type of convergence,
known as "uniform" convergence, which does guarantee that limit functions
inherit certain of these features from the functions in the sequence.

EXERCISE SET 9.1

1. Let S = [a, b] for some a < b. Which of the following are subspaces of
   B(S)? Verify your claim in each case.
   (a) The set of all f E B(S) that are piecewise-continuous on S ;
   (b) The set of all f E B(S) that are nonnegative on S;
   ( c) The set of all f E B ( S) that are analytic on S;
   (d) The set of all f E B(S) that are monotone increasing on S;
   (e) The set of all f E B(S) that are monotone on S;
   (f) The set of all f E B(S) such that f (xo) = 1 for some xo E S.


2. Verify the claims made in Example 9.1.7.