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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
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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Q5 "Real analysis "principal of Mathematical Analysis by rudin
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
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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