10 Show that (S) converges to a continuous function, but not uniformly. Use the series E f, to show that absolute convergence, even for all x, does not imply uni- form convergence.
10 Show that (S) converges to a continuous function, but not uniformly. Use the series E f, to show that absolute convergence, even for all x, does not imply uni- form convergence.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Q5 "Real analysis "principal of Mathematical Analysis by rudin
Transcribed Image Text:uniformly on E. If, in addition, {fa) and (g.) are sequences of bounded functions,
prove that {fag,} converges uniformly on E.
3. Construct sequences (S), {g.) which converge uniformly on some set E, but such
that (fag.) does not converge uniformly on E (of course, Sa9.) must converge on
E).
4. Consider
For what values of x does the series converge absolutely? On what intervals does
it converge uniformly? On what intervals does it fail to converge uniformly? Is f
continuous wherever the series converges? Is / bounded?
166 PRINCIPLES OF MATHEMATICAL ANALYSIS
5.
S(x) = sin
Show that (f) converges to a continuous function, but not uniformly. Use the
series E fa to show that absolute convergence, even for all x, does not imply uni-
form convergence.
6. Prove that the series
converges uniformly in every bounded interval, but does not converge absolutely
for any value of x.
7. For n=1, 2, 3, ..., x real, put
f.(x) -
1+ nx
Show that (f.) converges uniformly to a function f, and that the equation
S'(x) = lim f(x)
is correct if x # 0, but false if x= 0.
8. IГ
I(x) =
(xS0),
(x>0),
if (x,) is a sequence of distinct points of (a, b), and if Elc.l converges, prove that
the series
S(x) =
x- x.)
converges uniformly, and that fis continuous for every x + x,.
9. Let {fa} 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?
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
S(x) = (nx)
(x real).
Find all discontinuities of f, and show that they form a countable dense set.
Show that fis nevertheless Riemann-integrable on every bounded interval.
11 Sunnesa ff lal ane defined on F ond
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