9.3 Impli cations of Uniform Convergence in Calculus 5579.3 Implications of Uniform Convergence in
CalculusIt is natural to inquire about the interaction between limits of sequences of
functions and the fundamental operations of calculus such as limits, continuity,
derivatives, and integrals. To bring this concern into sharper focus, we pose the
following questions.FOUR QUESTIONS:
Suppose f n ---+ f pointwise on a set S S: JR and let xo E S.
Q#l If each function f n has a limit at xo, does f have a limit at x 0? If so, is it
true that lim ( lim f n(x)) = lim f(x)? That is, can we "interchange
n--+oo x--+xo x--+xothe limits," so that lim ( lim fn(x)) = lim ( lim fn(x))?
n--+oo x--+xo x--+x o n--+ooQ#2 If each function fn is continuous on S , must f also be continuous on S?
Q#3 If each function f n is integrable on [a, b] S: S, must f also be integrable
on [a, b]? If it is, must it be true that n--+oo lim t a f n = t a f?Q #4 If each function f n is differentiable at xo, must f also be differentiable at
x 0? If it is , must it be true that nlim --+oo f~(xo) = f'(xo)? ·
The following examples will show that the answer to each of these ques-
tions is "no." However, when "pointwise convergence" is replaced by "uniform
convergence,'' the answer to most of these questions changes to "yes." The
remainder of this section will be devoted to proving these claims.Examples 9.3.1(a) Define f n : [-1, l] ---+ JR
{-1 if - 1 < x < -.!. }
by Jn ( X) = 0 if - ~ : X : ~ n ·1 if .!. n < x < - 1
(See Figure 9.9.) The limit function is the _ 1
signum function defined in Definition 5.1.5.
Each function fn has a limit at 0, but f
does not. Thus, the answer to the first part
of Q#l is "no."yI I x
n n-1
Figure 9.9