Partition of an interval

A partition of a closed interval $[a,b]\subseteq\mathbb{R}$ is a finite set of points written in increasing order

The associated subintervals are $[x_{i-1},x_i]$ for $i=1,\dots,n$, and their lengths are $\Delta x_i:=x_i-x_{i-1}$.

Partitions are the indexing objects for Riemann sums , upper /lower sums , and the definition of the Riemann integral . See also refinement and mesh .

Examples:

• $P:\ 0<\tfrac12<1$ is a partition of $[0,1]$ with two subintervals.
• The uniform partition of $[0,1]$ into $n$ pieces is $x_i=i/n$.
• The trivial partition is $a=x_0<x_1=b$ (one subinterval).