Question: (1) Let's prove that there exist functions with infinitely many discontinuities which are (surprisingly)integrable.Let f:[0,1]Rbe defined byf(x)={1ifx=1mforsomeminN0otherwiseNotice that f has infinitely many discontinuities in this

(1) Let's prove that there exist functions with infinitely many discontinuities which are (surprisingly)integrable.Let f:[0,1]Rbe defined byf(x)={1ifx=1mforsomeminN0otherwiseNotice that f has infinitely many discontinuities in this interval.For NinN, set PNtobe the following partition of0,1 :{0,(1N+12N2),(1N-1-12N2),(1N-1+12N2),dots,(13-12N2),(13+12N2),(12-12N2),(12+12N2),(1-12N2),1}(a) For an integer k with 2kN, prove that every subinterval of the form(1k+12N2),(1k-1-12N2)contains no real numbers of the form x=1m, where misan integer.(b) Prove all the other types of subintervals; the ones of the form[(1k-12N2),(1k+12N2)],where 2kN-1as well as0,(1N+12N2)and(1-12N2),1do contain a real number of the form 1m for minN, as well as real numbers which are not ofthat form.(c) Use (a),(b)to compute U(f,PN) and L(f,PN).(d) Use the integrability criterion, together with help from (c),to prove that fis integrable on0,1.(e) Find 01f.

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

General proof tip: Before you start writing your proof, first work in a "scratchwork" section. Use your seratchwork to first figure out what you have, what you need and what you are going to do....

(4) Let f(2) be holomorphic on |2| 1 be a natural number. For any z with |2|

{Iz]

PROBlem 1 Let /BIZ1. Prove That + B+ 32 f . . . 3 Tket .

1. Let ECR. Prove that E is Lebesgue measurable if and only if it satisfies the Caratheodory's condition': For every ACR, it is true that |A| = |ANE|+|AN], where = R \ E.

Let L - D - A denote the unnormalized Laplacian matrix for an undirected, weighted graplh G(V,8), where A is a symmetric, nonnegative adjacency matrix, and D- diag[s1,..., sv] is a diagonal matrix...

1. Recursion: Define a Recursive Integer List (RIL) to be a list of a finite number of objects, where each object is either an integer or is an RIL. For example, L = (3, 2, [4, 3], [[3,9), (), 8], 7]...

Microkernel operating systems aim to address perceived modularity and reliability issues in traditional "monolithic" operating systems. (i) Describe the typical architecture of a microkernel...

15. Consider the mapping from M,(Z) into Z given by a b d -> a . Prove or disprove that this is a ring homomorphism. 16. Let R = 1. ab, cEZ . Prove or disprove that the map- ping a b C . - a is a...

Let n be an integer. Prove that 3n-11 is odd if and only if Let n be an integer. Prove that n is even if and only if n is even. Let n be an integer. Prove that n is odd if and only if n is odd. Let n...

Fixed costs are costs that: decrease as the per unit variable cost increases. All of these are correct. are constant within the relevant range as the level of output changes. increase as the...

Vassar Corp. has incurred substantial losses for several years and has decided to declare bankruptcy. The company petitioned the court for protection from creditors on March 31, 2010, and submitted...

Because of the fixed input nature of capital, _ _ _ _ _ _ _ _ _ _ is a feature of

Kashmir Enterprises, an Indian carpet manufacturer, begins the calendar year with the following Indian rupee (INR) balances: Cash 920,000 Accounts payable 420,000 Inventory 640,000 Owners equity...