Darboux sums of an indicator function.

Question

Let $a < c < d < b$. Let $f(x)=\begin{cases} 1 \text{ if } x \in [c,d]\\ 0 \text{ if } x \in [a,b]\setminus[c,d]. \end{cases}$

I want to verify that I understand how to properly apply the upper and lower Darboux sums.

Let $P=\{a=x_0<...<x_{k-1}<c<x_k<...<x_{l-1}<d<x_l<...<x_n=b\}$

$U(f,P)=\sum\limits_{i=1}^{n}M_i(f)\Delta x_i=\sum\limits_{i=k}^{l}M_i\Delta x_i=x_l-x_k \geq d-c.$

Verify that $L(f,P) \leq d-c$. This is my attempt at a verification.

$L(f,P)=\sum\limits_{i=1}^{n}m_i(f)\Delta x_i=\sum\limits_{i=k-1}^{l-1}m_i\Delta x_i=x_{l-1}-x_{k-1} \leq d-c.$

Is my verification correct?

Answer 1

Yes, this is correct except for some indices. Just to sum up: $$ M_i=\sup_{[x_{i-1},x_i]}f= \begin{cases}1,\quad &i=k,\ldots,l\\ 0,&\text{otherwise} \end{cases} $$ by the definition of $f$ and your partition $P$. A similar thing holds for $m_i$. Then $$ U(f,P)=\sum_{i=1}^n M_i(x_i-x_{i-1})=\sum_{i=k}^l (x_i-x_{i-1})=x_l-x_{k-1}\geq d-c, $$ and similarly for $L(f,P)$. Note that you want $x_l$ to be larger than $d$ but $x_{k-1}$ to be smaller than $c$ in order for the inequality to hold.