Question

A bounded function f:[a,b]->R is integrable on [a,b]
if there exists L in R such that for any
ϵ>0
, there exists
Pϵ
such that
|S(P,f)−L|<ϵ
for any partitions
P⊃Pϵ

Prove it!

Answer 1

good luck :)

Answer 2

help me to prove it @perl

Answer 3

well, i can try

Answer 4

what is L >

Answer 5

L is real number

Answer 6

\[L \in \mathbb{R} \]

Answer 7

What exactly are you trying to prove? That is just the definition of Riemann integrability.

Answer 8

im not sure

Answer 9

this is a definition, you dont prove a definition :)

Answer 10

i think he is trying to prove that a bounded function is integrable, oh

Answer 11

this is not definition, it's a theorem..

Answer 12

Definition : A bounded function f:[a,b] is integrable on [a,b] if there exists \[L \in \mathbb{R} \] such that for any \[\epsilon >0\] , there exists \[\delta >0\] such that for any partition P of [a,b] such that \[||P||>\delta \], we have \[|S(P,f)-L|<\epsilon \]

Answer 13

Something is missing here. x-1/2 is integrable between 0 and 1, but it is not bounded.

Answer 14

x^(-1/2)

Answer 15

ok so you want to prove that a bounded function is Riemann integrable?

Answer 16

there is also Darboux integrable

Answer 17

no, i want to prove that theorem that using refinement of partition of P..

Answer 18

I think its false. you can have an unbounded function that is integrable

Answer 19

this doesnt look like a theorem, it looks like a definition of a bounded integrable function

Answer 20

Okay, forget about that..
How to show that definition of Riemann integral and Darboux are equivalent?

Answer 21

this is just the definition of riemann integrable , or riemann criterion

Answer 22

Owh..thanks..:)
I have next question : Show that definition of Riemann Integral and Darboux are equivalent.