3. Let f(x) = 1/n for x = m/n with (m, n) = 1, and f(x) = 0 for irrational x. For irrationalu € (0, 1) let Fu(x) = 0 if x < u and F(x) = 1 if x ≥ u. Show that f is F-integrable on[0, 1].

The function f defined here assigns a value of 1n to rational numbers mn where m and n are coprime…

Answered: Let f(x) = 1/n for x = m/n with (m, n)…

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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A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1 then f is constant in [a, b).
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Let f : [0, 1] → R be a continuous function on the closed interval [0, 1] and differentiable on the open interval (0, 1). Assume that f(0) = 0 and f’ is an increasing function on (0, 1). Show that g(x) =f(x)/x is an increasing function on (0, 1)
Suppose that f: R2 → R is a function given by S x² + y² f(x, y) = 0 for x = y = 0. Show that f(x, y) is continuous at point (0, 0). for (x, y) = (0,0),
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1. Let f(x) be a decreasing function on [0, 1]. Show that f is integrable over [0, 1].
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Let C = sin dx. As f(x) = sin(1/x) is continuous except at 0 and bounded, ƒ is integrable on [-1, 1]. Since the integrand is positive (never negative), C > 0. Put u = varibles to u so that f(x) = sin(u)² and dx = -u-2du. When x = u = -1. Therefore, 1/x and make the change of = 1, u = 1 and when x = -1, ´sin(u) = = - sin(u) (u-2)du C = du < 0. и Where does this argument go wrong?

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Let f(x) = 1/n for x = m/n with (m, n) = 1, and f(x) = 0 for irrational x. For irrational u € (0, 1) let F(x) = 0 if x < u and F(x) = 1 if r2u. Show that f is F-integrable on [0, 1].