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Partition of an interval

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A partition of an interval is a division of an interval into several disjoint sub-intervals. Partitions of intervals arise in calculus in the context of Riemann integrals.

Contents

Definition

Let [a,b] be an interval of real numbers.

A partition \mathcal{P} is defined as the ordered n-tuple of real numbers \mathcal{P}=(x_0,x_1,\ldots,x_n) such that a=x_0<x_1<\ldots<x_n=b

Norm

The norm of a partition \mathcal{P} is defined as \|\mathcal{P}\|=\sup\{x_i-x_{i-1}\}_{i=1}^n

Tags

Let \mathcal{P}=\{x_0,x_1,\ldots,x_n\} be a partition.

A Tagged partition \mathcal{\dot{P}} is defined as the set of ordered pairs \mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n.

Where x_{i-1}<t_i<x_i\forall t_i. The points t_i are called the Tags.

See also

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