# Definition:Subdivision (Real Analysis)/Finite

< Definition:Subdivision (Real Analysis)(Redirected from Definition:Finite Subdivision)

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## Definition

Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers.

Let $x_0, x_1, x_2, \ldots, x_{n - 1}, x_n$ be points of $\R$ such that:

- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$

Then $\set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a **finite subdivision of $\closedint a b$**.

### Normal Subdivision

$P$ is a **normal subdivision of $\closedint a b$** if and only if:

- the length of every interval of the form $\closedint {x_i} {x_{i + 1} }$ is the same as every other.

That is, if and only if:

- $\exists c \in \R_{> 0}: \forall i \in \N_{< n}: x_{i + 1} - x_i = c$

## Also known as

Some sources use the term **partition** for the concept of a subdivision.

However, the latter term has a different and more general definition, so its use is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources do not define the concept of infinite subdivision, and so simply refer to a **finite subdivision** as just a **subdivision**.

## Sources

- 1970: Arne Broman:
*Introduction to Partial Differential Equations*... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions - 1973: Tom M. Apostol:
*Mathematical Analysis*(2nd ed.) ... (previous) ... (next): $\S 6.3$: Functions of Bounded Variation: Definition $6.3$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 13.2$