Problem 4 (4 points each) Practice with pointwise and uniform convergence. (a) Let fn: (0, 1) → R be given by fn(x) = n+1. Show that {f} converges pointwise to a continuous function f, but the convergence is not uniform. nx (Remark: This shows that pointwise convergence to a continuous function does not imply uniform convergence, so the "converse" to Theorem 6.2.2 is not true. It is also possible to find counterexamples using sequences of continuous functions on [0, 1]) (b) Let fn [0, 1] → R be defined by fn(2): = x = 0 n=0 < x≤ 1/1/0 0 < x < 1 Notice that fn € R[0, 1] since it has a finite number of discontinuities. Show that {fn} converges pointwise to function f € R[0, 1], but the convergence is not uniform (without using Theorem 6.2.4). Furthermore, show that lim n→∞ [² sn + [² s fn f (c) Let fn(x) = . Show that {f} converges uniformly to a differentiable function f on [0, 1] (find f). However, show that ƒ'(1) ‡ lim f(1). n→∞
Problem 4 (4 points each) Practice with pointwise and uniform convergence. (a) Let fn: (0, 1) → R be given by fn(x) = n+1. Show that {f} converges pointwise to a continuous function f, but the convergence is not uniform. nx (Remark: This shows that pointwise convergence to a continuous function does not imply uniform convergence, so the "converse" to Theorem 6.2.2 is not true. It is also possible to find counterexamples using sequences of continuous functions on [0, 1]) (b) Let fn [0, 1] → R be defined by fn(2): = x = 0 n=0 < x≤ 1/1/0 0 < x < 1 Notice that fn € R[0, 1] since it has a finite number of discontinuities. Show that {fn} converges pointwise to function f € R[0, 1], but the convergence is not uniform (without using Theorem 6.2.4). Furthermore, show that lim n→∞ [² sn + [² s fn f (c) Let fn(x) = . Show that {f} converges uniformly to a differentiable function f on [0, 1] (find f). However, show that ƒ'(1) ‡ lim f(1). n→∞
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
![Problem 4 (4 points each) Practice with pointwise and uniform convergence.
(a) Let fn: (0, 1) → R be given by fn(x) = n+1. Show that {f} converges pointwise to
a continuous function f, but the convergence is not uniform.
nx
(Remark: This shows that pointwise convergence to a continuous function does not
imply uniform convergence, so the "converse" to Theorem 6.2.2 is not true. It is also
possible to find counterexamples using sequences of continuous functions on [0, 1])
(b) Let fn [0, 1] → R be defined by
fn(2):
=
x = 0
n=0 < x≤ 1/1/0
0 < x < 1
Notice that fn € R[0, 1] since it has a finite number of discontinuities. Show that
{fn} converges pointwise to function f € R[0, 1], but the convergence is not uniform
(without using Theorem 6.2.4). Furthermore, show that
lim
n→∞
[² sn + [² s
fn
f
(c) Let fn(x) = . Show that {f} converges uniformly to a differentiable function f on
[0, 1] (find f). However, show that ƒ'(1) ‡ lim f(1).
n→∞](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F679e8689-a8d0-4611-943b-8f3d73af0804%2F56145509-2b6b-4838-bdb6-382e11a4bc1e%2F5dzw8me_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 4 (4 points each) Practice with pointwise and uniform convergence.
(a) Let fn: (0, 1) → R be given by fn(x) = n+1. Show that {f} converges pointwise to
a continuous function f, but the convergence is not uniform.
nx
(Remark: This shows that pointwise convergence to a continuous function does not
imply uniform convergence, so the "converse" to Theorem 6.2.2 is not true. It is also
possible to find counterexamples using sequences of continuous functions on [0, 1])
(b) Let fn [0, 1] → R be defined by
fn(2):
=
x = 0
n=0 < x≤ 1/1/0
0 < x < 1
Notice that fn € R[0, 1] since it has a finite number of discontinuities. Show that
{fn} converges pointwise to function f € R[0, 1], but the convergence is not uniform
(without using Theorem 6.2.4). Furthermore, show that
lim
n→∞
[² sn + [² s
fn
f
(c) Let fn(x) = . Show that {f} converges uniformly to a differentiable function f on
[0, 1] (find f). However, show that ƒ'(1) ‡ lim f(1).
n→∞
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
