Hello everyone. I'm studying Darboux/Riemann sums with its lower and upper integrals.
Suppose a function $f\colon [a,b]\to \mathbb{R}$ and a partition $P=\{t_0,t_1,\ldots,t_n\}$ with $a=t_0 < t_1 < \cdots < t_n = b$.
The book proposes a partition refinement $Q = \{t_0,\ldots, t_{i-1}, r, t_{i}, \ldots, t_n\}$. That is, we are adding a new point $r\in Q$.
We define $m_i = \inf\{f(x):x\in [t_{i-1},t_i]\}$, $M_i = \sup\{f(x):x\in [t_{i-1},t_i]\}$ and
\begin{align*}
s(f;P)&=m_1 (t_1-t_0)+\cdots+m_n (t_n-t_{n-1})\\
Suppose a function $f\colon [a,b]\to \mathbb{R}$ and a partition $P=\{t_0,t_1,\ldots,t_n\}$ with $a=t_0 < t_1 < \cdots < t_n = b$.
The book proposes a partition refinement $Q = \{t_0,\ldots, t_{i-1}, r, t_{i}, \ldots, t_n\}$. That is, we are adding a new point $r\in Q$.
We define $m_i = \inf\{f(x):x\in [t_{i-1},t_i]\}$, $M_i = \sup\{f(x):x\in [t_{i-1},t_i]\}$ and
\begin{align*}
s(f;P)&=m_1 (t_1-t_0)+\cdots+m_n (t_n-t_{n-1})\\
