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Difference between revisions of "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 integral]]s.
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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 [[Integral#Riemann Integral|Riemann integral]]s.
  
 
==Definition==
 
==Definition==

Latest revision as of 20:34, 6 March 2022

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.

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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