Interval Partitions

Why

We partition a real interval into interval pieces.

Definition

An interval partition is a finite partition of a closed real interval.

An interval partition is regular if all pieces except the largest are closed on the left and open on the right and the largest is closed.

Any regular interval partition with $n-1$ elements can be represented by $n+1$ real numbers: the endpoints of each interval. We call these the cut points of the interval partition.

Notation

Let $[a, b]$ a closed interval in $\R $ with endpoints $a, b \in \R $.

Consider a regular partition of $[a,b]$ with $n-1$ pieces. We can identify its cut points:

\[ a = a_1 < a_2 < \dots a_{n-1} < a_n = b. \]