Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

9.4.E: Problems on Uniform Convergence of Functions and C-Integrals

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 9.4.E.1

Fill in all proof details in Theorems 15, Corollaries 4 and 5, and examples (A) and (B).

Exercise 9.4.E.1

Using (6), prove that
limxqH(x,y) (uniformly) 


exists on BE1 iff
(ε>0)(G¬q)(yB)(x,xG¬q)|H(x,y)H(x,y)|<ε.

Assume E complete and |H|< on G¬q×B.
[Hint: "Imitate" the proof of Theorem 1, using Theorem 2 of Chapter 4, §2.]

Exercise 9.4.E.2

State formulas analogous to ( 1) and ( 2) for a,ba, and ba+.

Exercise 9.4.E.3

State and prove Theorems 1 to 3 and Corollaries 1 to 3 for
a,ba, and ba+.


In Theorems 2 and 3 explore absolute convergence for
ba and ba+.

Do at least some of the cases involved.
[Hint: Use Theorem 1 of §3 and Problem 1', if already solved.]

Exercise 9.4.E.4

Prove that
limxqH(x,y)=F(y) (uniformly) 


on B iff
limnH(xn,)=F( uniformly )

on B for all sequences xnq(xnq).
[Hint: "Imitate" Theorem 1 in Chapter 4, §2. Use Definition 1 of Chapter 4, §12.]

Exercise 9.4.E.5

Prove that if
limxqH(x,y)=F(y) (uniformly) 


on A and on B, then this convergence holds on AB. Hence deduce similar propositions on C-integrals.

Exercise 9.4.E.6

Show that the integrals listed below violate Corollary 4 and hence do not converge uniformly on P=(0,δ) though proper L-integrals exist for each uP. Thus show that Theorem 1 (ii) does not apply to uniform convergence.
(a) 10+udtt2u2;
(b) 10+u2t2(t2+u2)2dt;
(c) 10+tu(t2u2)(t2+u2)2dt.
[ Hint for (b): To disprove uniform convergence, fix any ε,v>0. Then
v0u2t2(t2+u2)2dt=vv2+u21v


as u0. Thus if v<12ε,
(uP)v0u2t2(t2+u2)2dt>12v>ε.]

Exercise 9.4.E.7

Using Corollaries 3 to 5, show that the following integrals converge (uniformly) on U (as listed) but only pointwise on P (for the latter, proceed as in Problem 6 ). Specify P and M(t) in each case where they are not given.
(a) 0eut2dt;U=[δ,);P=(0,δ).
[ Hint: Set M(t)=eδt for t1 (Corollaries 3 and 5).]
(b) 0euttacostdt(a0);U=[δ,).
(c) 10+tu1dt;U=[δ,).
(d) 10+tusintdt;U=[0,δ],0<δ<2;P=[δ,2);M(t)=t1δ.
[Hint: Fix v so small that
(t(0,v))sintt>12.


Then, if u2,
v0tusintdt12v0dttu1.]

Exercise 9.4.E.8

In example (A), disprove uniform convergence on P=(0,).
[Hint: Proceed as in Problem 6.]

Exercise 9.4.E.9

Do example (B) using Theorem 3 and Corollary 5. Disprove uniform convergence on B.

Exercise 9.4.E.10

Show that
0+sintutcostdt


converges uniformly on any closed interval U, with ±1U.
[Hint: Transform into
120+1t{sin[(u+1)t]+sin[(u1)t]}dt.]

Exercise 9.4.E.11

Show that
0tsint3sintudt


converges (uniformly) on any finite interval U.
[Hint: Integrate
yxtsint3sintudt

by parts twice. Then let y and x0.]

Exercise 9.4.E.12

Show that
0+etucosttadt(0<a<1)


converges (uniformly) for u0.
[Hints: For t0+, use M(t)=ta. For t, use example (B) and Theorem 2. ]

Exercise 9.4.E.13

Prove that
0+costutadt(0<a<1)


converges (uniformly) for uδ>0, but (pointwise) for u>0.
[Hint: Use Theorem 3 with g(t,u)=costu and
|x0g|=|sinxuu|1δ.

For u>0,
vcostutadt=ua1vucoszzdz

if v=1/u and u0. Use Corollary 4.]

Exercise 9.4.E.14

Given A,BE1(mA<) and f:E2E, suppose that
(i) each f(x,)=fx(xA) is relatively (or uniformly) continuous on B; and
(ii) eachf(,y)=fy(yB) is m-integrable on A.
Set
F(y)=Af(x,y)dm(x),yB.


Then show that F is relatively (or uniformly) continuous on B.
[Hint: We have
(xA)(ε>0)(y0B)(δ>0)(yBGy0(δ))|F(y)F(y0)|A|f(x,y)f(x,y0)|dm(x)A(εmA)dm=ε.

Similarly for uniform continuity.]

Exercise 9.4.E.15

Suppose that
(a) Caf(t,y)dm(t)=F(y)( uniformly ) on B=[b,d]E1;
(b) each f(x,)=fx(xa) is relatively continuous on B; and
(c) each f(,y)=fy(yB) is m -integrable on every [a,x]E1, xa.
Then show that F is relatively continuous, hence integrable, on B and that
BF=limxBHx,


where
H(x,y)=xaf(t,y)dm(t).

(Passage to the limit under the -sign.)
[Hint: Use Problem 14 and Theorem 4; note that
C0f(t,y)dm(t)=limxH(x,y)(uniformly).]


9.4.E: Problems on Uniform Convergence of Functions and C-Integrals is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

Support Center

How can we help?