Problem 4: Define f: [0, 1] R by f(x) = 2x if x is irrational and f(x) =-1 if x is rational. Carefully demonstrate, using Definition 7.1.3, that f is not Riemann-integrable on [0,1]. ww.
Problem 4: Define f: [0, 1] R by f(x) = 2x if x is irrational and f(x) =-1 if x is rational. Carefully demonstrate, using Definition 7.1.3, that f is not Riemann-integrable on [0,1]. ww.
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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![Problem 4: Define f: [0, 1] → R by f(x) = 2x if x is irrational and f(x) = -1 if x is
rational. Carefully demonstrate, using Definition 7.1.3, that f is not Riemann-integrable on
[0, 1].
%3D
Solution:
7.1.3 DEFINITION
Let f be a bounded function defined on [a, b]. Then
U(f) = inf {U(fS, P): P is a partition of [a, b]}
is called the upper integral of f on [a, b]. Similarly,
L(f) = sup {L(f. P): P is a partition of [a, b]}
%3D
is called the lower integral of f on [a, b). If these upper and lower integrals
are equal, then we say that f is Riemann integrable on [a, b], and we denote
their common value by f or by LSx) dx. That is, if L(f)= U(f), then
S-Lrmdx = LUS) = U(S)
is the Riemann integral of f on [a, b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7e7c908-505f-4c35-8ac4-e541f2ab69b2%2F5a0d7152-7ed9-47e8-b571-1d59227d8320%2Fnhoxjim_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4: Define f: [0, 1] → R by f(x) = 2x if x is irrational and f(x) = -1 if x is
rational. Carefully demonstrate, using Definition 7.1.3, that f is not Riemann-integrable on
[0, 1].
%3D
Solution:
7.1.3 DEFINITION
Let f be a bounded function defined on [a, b]. Then
U(f) = inf {U(fS, P): P is a partition of [a, b]}
is called the upper integral of f on [a, b]. Similarly,
L(f) = sup {L(f. P): P is a partition of [a, b]}
%3D
is called the lower integral of f on [a, b). If these upper and lower integrals
are equal, then we say that f is Riemann integrable on [a, b], and we denote
their common value by f or by LSx) dx. That is, if L(f)= U(f), then
S-Lrmdx = LUS) = U(S)
is the Riemann integral of f on [a, b].
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