Purchase Solution

Riemann integrable function on [0, 1]

Not what you're looking for?

Ask Custom Question

Consider a function fâ?¶[0,1]â?'[0,1] given by
f(x)={(1/q, if x=p/q,where p,qâ??N are coprime,
0, if x is irrational,
1, if x=0.

(a) Show that if xâ??[0,1] is rational, then f is not continuous at x.

(b) Show that L(f,P)=0 for any partition P of [0,1].

(c) Show that for any qâ??N the number of elements in the set
X(q)={xâ??[0,1]:f(x)â?¥1/q }
can be bounded by q(q+3)�2. (Better bounds are possible, but not required.)

(d) Using that X(q) is finite, show that f is continuous at any irrational xâ??[0,1].

(e)Show that for any ?>0, we can find a partition P_? of [0,1] such that U(f,P_?)<?.

(f)Deduce that f is Riemann integrable and
â?«^1_0 f(x)dx=0.

Attachments
Purchase this Solution
Solution Summary

A complete, detailed solution is provided.

Solution provided by:
Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
Recent Feedback
  • "Thanks for your assistance. "
  • "Thank you. I understand now."
  • "Super - Thank You"
  • "Very clear. I appreciate your help. Thank you."
  • "Great. thank you so much!"
Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Probability Quiz

Some questions on probability

View More Free Quizzes
Related BrainMass Solutions

  • Lebesgue Integration

    If a=0, f(x)=1 is obviously not integrable 2. if a>0, the function is Riemann-integrable in every interval [-R,R].

  • Riemann integration

    The integral of f(x) from -1 to 1 is 4. So f(x) is Riemann integrable. 2. f is continuous on [a,b]. This is not necessary. Here is an example. Let f(x)=0 on [0,1) and f(x)=1 on [1,2]. Then the integral of f(x) from 0 to 2 is 1.

  • Real Analysis : Riemann Integrable Functions

    103513 Real Analysis : Riemann Integrable Functions Show that the function h, defined on I by h(x)=x for x rational and h(x)=0 for x irrational, is not Riemann integrable on I. Please see the attached file for the complete solution.

  • Ruler function proof

    Proving Ruler Function is Riemann Integrable without using Riemann-Lebesgue Theorem Define Ruler Function g:(0,1)-> R (it is the usual ruler function available online) as g(x) = { 1/q if x = p/q, p and q are relatively prime { 0

  • Riemann Sum

    30747 Riemann Sum Investigated Q. Show directly that the function {see attachment}, is integrable on R = [0,1] x [0,1] and find {see attachment} (Hint: Partition R into {see attachment} squares and let N ...

  • Real Analysis: Riemann Integrability

    9711 Real Analysis: Riemann Integrability Please see the attached file for the fully formatted problem. Let E = { 1/n : n 2 N } . Prove that the function f(x) = ? 1 x 2 E 0 otherwise is integrable on [0,1].

  • Real Analysis : Integrable Functions

    158573 Real Analysis: Integrable Functions Let f:[0,1]-->R be a Riemann integrable function. Prove that lim n-->∞ ∫ (from 0 to 1) x^n f(x)dx = 0. I do not know where to begin on this problem. It seems like it should be easy though.

  • Riemand integrable

    449713 Riemand integrable prove if f(x)=1 if x= 1/n ,n is positive integer f(x)= 0 else, is Riemand integrable on [0,1] Define a. Prove that f is integrable on [0, 1] . b.

  • Riemann Stieltjes Integration

    Hence, we conclude that f is NOT Riemann Stieltjes Integrable. NOTE: If f is Riemann Stieltjes Integrable, then for every epsilon>0, for any partition P of [a,b] and any choices of s_i and t_i, U(f,P)-L(f,P)

  • Example of Lebesgue Measure and Lebesgue Integration

    Let f be the following function with domain C = [0, 1] X [0, 1] (in two-dimensional Cartesian space): f(x, y) = 0 on the line segments x = 0, y = 0, and x = y f(x, y) = -1/(x^2) if 0 < y < x <= 1 f(x, y) = 1/(y^2) if 0 < x < y <= 1 Compute

View More Related Solutions