Partition of an Interval¶

Definition¶

A partition of $[a,b]$ is a finite ordered set

P=\{x_0,x_1,\ldots,x_n\}\quad \text{with}\quad a=x_0<x_1<\cdots<x_n=b.

Its subintervals are $J_i=[x_{i-1},x_i]$ for $1\leq i\leq n\text{,}$ and its mesh is

\lVert P\rVert=\max_{1\leq i\leq n}(x_i-x_{i-1}).

If all subintervals have equal length, then $P$ is a regular subdivision.

Interpretation¶

A partition cuts $[a,b]$ into finitely many consecutive subintervals at the points $a=x_0<x_1<\cdots<x_n=b$ $b$ . It is the bookkeeping behind every Riemann sum: the function is approximated separately on each subinterval and the pieces are added up. A regular subdivision is the simplest choice, with all pieces of the same length.

Two copies of an interval from a to b: a partition into unequal pieces by points x0 to x6, and a regular subdivision into n equal pieces of width delta x.

The upper line is a partition with $n=6$ pieces of different lengths; its mesh is the length of the widest piece, here $[x_2,x_3]$ . The lower line is a regular subdivision with points $a+i\,\Delta x$ $i Δ x$ : every piece has length $\Delta x=\tfrac{b-a}{n}$ $\frac{b - a}{n}$ , and this is also its mesh.

The conditions in the definition make the pieces fit together. The points increase strictly, so every piece has positive length, and the first and last points are $a$ and $b$ , so the pieces cover $[a,b]$ and overlap only at shared endpoints. The mesh $\lVert P\rVert$ measures fineness because it controls all pieces at once; many points alone are not enough. If $x_1=\tfrac{a+b}{2}$ and all further points lie in the right half, then $\lVert P\rVert\ge\tfrac{b-a}{2}$ however large $n$ is. For a regular subdivision the two notions agree, since $\lVert P\rVert=\tfrac{b-a}{n}\to0$ exactly when $n\to\infty$ $\infty$ . This is why limits over regular subdivisions, as for the Riemann integral, are taken as $n\to\infty$ $\infty$ .

A common slip is to identify $P$ with its subintervals. $P$ is the set of cut points, so a partition into $n$ pieces has $n+1$ points, and neighbouring subintervals $[x_{i-1},x_i]$ and $[x_i,x_{i+1}]$ share the point $x_i$ .

Key Results¶

Length is finitely additive: for every partition $P$ of $[a,b]\text{,}$ $\displaystyle b-a=\sum_{i=1}^n (x_i-x_{i-1})\text{.}$
Every closed interval $[a,b]$ with $a<b$ admits partitions of arbitrarily small mesh.
Regular subdivisions are special cases of partitions; the regular subdivision of $[a,b]$ into $n$ pieces has mesh $(b-a)/n\text{.}$

Examples¶

Dividing $[0,1]$ into $n$ equal subintervals forms a regular subdivision.
Irregular partitions allow variable subinterval lengths.

Prerequisites¶

Length of an Interval

Sources¶

Math I: Calculus · Lecture 10

Aliases: partition of a closed interval Keywords: regular subdivision, mesh, mesh of a partition, subinterval Symbols: delta, infinity, sum

Last updated: 27 October 2026 11:52