Riemann integral 2

date: 2025-01-08

Example 1

Dirichlet fn. defined by

Clearly and Pick a partition Now =0 and

Example

Let

Example

but

(::todo)

Definition

Let we say or is a refinement of in nodes of nodes of

Proposition

Let suppose Then

Enough to prove Set with set let let

Exercise

Complete the exercies for U(, ) and induction

Corollary

Let then

Corollary

Theorem

Let then

Theorem

Let then for $ε ε$

Let $ε$ $ε$

Now $ε ε$

ε

Let $ε$ $ε$ $ε$

$ε ε ε$

Definition

Let define

Darboux

let then for $ε ε$ with

Definition

by

is trivial so let’s do Let $ε$ $ε$

Let # nodes set $ε$ pick and assume set

has at most point that are not in Let

$ε$

$ε ε ε$