- #1
snipez90
- 1,101
- 5
Homework Statement
Let f(x) = 0, if x is rational; 2 + x, if x is irrational, for all x in [0,1]. Show that f is not integrable
Homework Equations
density of rationals/irrationals, equation for L(f,P) and U(f,P), Darboux Integrability criterion
The Attempt at a Solution
L(f,P) = 0 because on any subinterval formed from two consecutive points in the partition of [0,1], there exists a rational x.
For U(f,P), I was thinking that I could just pick any irrational x so that sup{ f(x): x is an element of the subinterval from two consecutive points in the partition of [0,1] } = 2 + x. Then note that since [tex]2+x \geq 2[/tex], [tex]U(f,P) \geq 2(b-a) > 0[/tex] so that
[tex]sup {L(f,P)} \neq inf {U(f,P)}[/tex]
I left a few steps in demonstrating U(f,P) > 0 but is this a good approach?