# Give an expression for lower and upper sum, concluding $f$ is integrable

$f : [a,b] \rightarrow \mathbb{R}$ is non-increasing, which means that $f(y) \le f(x)$ when $y>x$

The two questions are :

1. Give an explicit expression (without infima and suprema) of $L(f,P)$ and $U(f,P)$ for any partition $P$ of $[a,b]$

2. Prove that the function is integrable on $[a,b]$

Definitions:

A Partition P of [a,b] is a finite, ordered set

$$P = \text{ { a = x_0<x_1<x_2< ... < x_n =b}}$$ For each subinterval $[x_{k-1}, x_k]$ of P, let

$$m_k= \inf (f(x): x \in [x_{k-1}, x_k] ) \text{ and } M_k= \sup (f(x): x \in [x_{k-1}, x_k] )$$

The Lower Sum of $f$ with respect to $P$ is given by $$L(f,P)= \sum_{k=1}^n m_k(x_k - x_{k-1})$$ The Upper Sum of $f$ with respect to $P$ is given by $$U(f,P)= \sum_{k=1}^n M_k(x_k - x_{k-1})$$

For part 2) of this question I need the following theorem:

A bounded function $f$ is integrable on $[a,b]$ if and only if, for every $\epsilon>0$, there exists a Partition $P_\epsilon$ of $[a,b]$ such that $$U(f,P_\epsilon)-L(f, P_\epsilon) < \epsilon$$

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1. Is just asking what is $\sup \{f(x) : x \in [x_{k-1}, x_k]\}$ and the same for $\inf$, knowing that $f$ does not increase. Can you do that? If not, try drawing a graph of a non-increasing function. –  Karolis Juodelė Oct 23 '12 at 16:37
Hmm. I have to give an explicit expression without infima and suprema of L(f,P) and U(f,P)... –  MSKfdaswplwq Oct 23 '12 at 17:05
Well, if $f$ is non-increasing, you know that $\sup_{x\in [c,d]} = f(c)$... –  copper.hat Oct 23 '12 at 17:09
Given that $f$ is non-increasing, $\inf$ and $\sup$ of $f([a, b])$ are obvious. Try drawing it. –  Karolis Juodelė Oct 23 '12 at 17:09
So inf{f(x):x∈[xk−1,xk]}= sup{f(x):x∈[xk−1,xk]} ? So part2.) is that U - L = 0? –  MSKfdaswplwq Oct 23 '12 at 18:19
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