converges uniformly, and that f is continuous for every x + x. 9. Let (f) be a sequence of continuous functions which converges uniformly to a function f on a set E. Prove that lim f.(x.) =f(x) for every sequence of points x, e E such that x, -+ x, and x e E. Is the converse of this true?
converges uniformly, and that f is continuous for every x + x. 9. Let (f) be a sequence of continuous functions which converges uniformly to a function f on a set E. Prove that lim f.(x.) =f(x) for every sequence of points x, e E such that x, -+ x, and x e E. Is the converse of this true?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Q9 real analysis by Walton rudin
![12:01 ..l il O
42
Rudin principal.
(xSU),
(x>0),
I(x) =
if (xn) is a sequence of distinct points of (a, b), and if E c. converges, prove that
the series
f(x) =
È c, (x – x»)
(asxsb)
converges uniformly, and that f is continuous for every x + x,.
9. Let {f.} be a sequence of continuous functions which converges uniformly to a
function f on a set E. Prove that
lim fa(x,) = f(x)
for every sequence of points x, e E such that x, ->x, and x € E, Is the converse of
this true?
SEQUENCES AND SERIES OF FUNCTIONS 167
10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4,
for the definition), consider the function
(пх)
S(«) =
(x real).
%3D
Find all discontinuities of f, and show that they form a countable dense set.
Show that f is nevertheless Riemann-integrable on every bounded interval.
11. Suppose {L), {e) are defined on E, and
(a) Ef, has uniformly bounded partial sums;
(b) g.-0 uniformly on E;
(c) g.(x)29:(x)Z9(x)2 for every x e E.
Prove that E f.g, converges uniformly on E. Hint: Compare with Theorem
3.42.
12. Suppose g and f.(n= 1, 2, 3, ...) are defined on (0, o), are Riemann-integrable on
[t, T] whenever 0<t<T<o, |fal <9, f.+f uniformly on every compact sub-
set of (0, 0), and
g(x) dx < o.
Prove that
lim
, S(x) dx =
(See Exercises 7 and 8 of Chap. 6 for the relevant definitions.)
This is a rather weak form of Lebesgue's dominated convergence theorem
(Theorem 11.32). Even in the context of the Riemann integral, uniform conver-
gence can be replaced by pointwise convergence if it is assumed that fe R. (See
the articles by F. Cunningham in Math. Mag., vol. 40, 1967, pp. 179–186, and
by H. Kestelman in Amer. Math. Monthly, vol. 77, 1970, pp. 182–187.)
13. Assume that {f.) is a sequence of monotonically increasing functions c
OSf.(x)<1 for all x and all n.
(a) Prove that there is a function f and a sequence {n} such that
f(x) = lim f,(x)
TL. aintaan f uah nintuine aonuannant hnaauannn in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa29dd1fd-ff33-4092-aedd-03e5d354d4ee%2F86a62213-2f05-4b3f-8d1f-a00f1be6546e%2Fpyjfdjd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:12:01 ..l il O
42
Rudin principal.
(xSU),
(x>0),
I(x) =
if (xn) is a sequence of distinct points of (a, b), and if E c. converges, prove that
the series
f(x) =
È c, (x – x»)
(asxsb)
converges uniformly, and that f is continuous for every x + x,.
9. Let {f.} be a sequence of continuous functions which converges uniformly to a
function f on a set E. Prove that
lim fa(x,) = f(x)
for every sequence of points x, e E such that x, ->x, and x € E, Is the converse of
this true?
SEQUENCES AND SERIES OF FUNCTIONS 167
10. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4,
for the definition), consider the function
(пх)
S(«) =
(x real).
%3D
Find all discontinuities of f, and show that they form a countable dense set.
Show that f is nevertheless Riemann-integrable on every bounded interval.
11. Suppose {L), {e) are defined on E, and
(a) Ef, has uniformly bounded partial sums;
(b) g.-0 uniformly on E;
(c) g.(x)29:(x)Z9(x)2 for every x e E.
Prove that E f.g, converges uniformly on E. Hint: Compare with Theorem
3.42.
12. Suppose g and f.(n= 1, 2, 3, ...) are defined on (0, o), are Riemann-integrable on
[t, T] whenever 0<t<T<o, |fal <9, f.+f uniformly on every compact sub-
set of (0, 0), and
g(x) dx < o.
Prove that
lim
, S(x) dx =
(See Exercises 7 and 8 of Chap. 6 for the relevant definitions.)
This is a rather weak form of Lebesgue's dominated convergence theorem
(Theorem 11.32). Even in the context of the Riemann integral, uniform conver-
gence can be replaced by pointwise convergence if it is assumed that fe R. (See
the articles by F. Cunningham in Math. Mag., vol. 40, 1967, pp. 179–186, and
by H. Kestelman in Amer. Math. Monthly, vol. 77, 1970, pp. 182–187.)
13. Assume that {f.) is a sequence of monotonically increasing functions c
OSf.(x)<1 for all x and all n.
(a) Prove that there is a function f and a sequence {n} such that
f(x) = lim f,(x)
TL. aintaan f uah nintuine aonuannant hnaauannn in
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