Let \(f\) be a function defined on the interval \([a,b]\text{,}\) and let \(n\) be a positive integer. A partition of \([a,b]\) into \(n\) subintervals is a choice of points \(x_0, x_1,\dots,
x_n\) satisfying
\begin{equation}
a=x_0\lt x_1\lt x_2\lt \cdots \lt x_n=b\text{.}\tag{1.3.3}
\end{equation}
Given a partition
(1.3.3) and integer
\(1\leq k\leq n\) the
\(k\)-th subinterval \(I_k\) is the interval
\([x_{k-1},x_k]\text{.}\) We denote by
\(\Delta x_k\) the
length of the subinterval
\(I_k\text{.}\) Thus we have
\begin{equation}
\Delta x_k=x_{k}-x_{k-1}\text{.}\tag{1.3.4}
\end{equation}
The norm of a partition \(P\text{,}\) denoted \(\norm{P}\text{,}\) is the maximum length of a subinterval: i.e.,
\begin{equation*}
\norm{P}=\max\{\Delta x_1, \Delta x_2, \dots, \Delta x_n\}\text{.}
\end{equation*}
Given a choice of sample points (or sample inputs) \(c_k\in I_k\) for each subinterval, the corresponding Riemann sum of \(f\) on \([a,b]\) is
\begin{equation*}
\sum_{k=1}^n f(c_k)\Delta x_k=f(c_1)(x_1-x_0)+f(c_2)(x_2-x_1)+\cdots +f(c_n)(x_n-x_{n-1})\text{.}
\end{equation*}
A partition
(1.3.3) along with a choice of sample points
\(c_k\in I_k\) for each
\(1\leq k\leq n\) is called a
pointed partition of
\([a,b]\text{.}\) In general we get a different Riemann sum of
\(f\) for each pointed partition of the interval
\([a,b]\) we choose.
