Intervals

Overview

An interval corresponds to a continuous segment of the real number line. There are a few different types. For all $a,b\in \mathbb{R}$ satisfying $a<b$:

• $[a,b]$ denotes a closed interval, all $x$ satisfying $a\le x\le b$;
• $(a,b)$ denotes an open interval, all $x$ satisfying $a<x<b$;
• $(a,b]$ denotes a half-open interval, all $x$ satisfying $a<x\le b$;
• $[a,b)$ denotes a half-open interval, all $x$ satisfying $a\le x<b$.

Partitions

Let $a,b\in \mathbb{R}$ such that $a<b$. A partition $P$ of interval $[a,b]$ is a set of points $x_0=a,x_1,\dots ,x_{n-1},x_n=b$ satisfying

We use the symbol $P=\{x_0,x_1,\dots ,x_n\}$ to designate this partition.

A refinement $P^{\prime }$ of some partition $P$ is created by adjoining more subdivision points to those of $P$. $P^{\prime }$, also a partition, is said to be finer than $P$. Given two partitions $P_1$ and $P_2$, the common refinement of $P_1$ and $P_2$ is the partition formed by adjoining the subdivision points of $P_1$ and $P_2$ together.

A function $f$ is said to be piecewise monotonic on an interval $[a,b]$ if there is a partition $P$ of $[a,b]$ such that $f$ is monotonic on each open subinterval of $P$.

Step Functions

A function $s$, whose domain is a closed interval $[a,b]$, is called a step function if and only if there exists a partition $P=\{a,x_1,\dots ,x_{n-1},b\}$ of $[a,b]$ such that $s$ is constant on each open subinterval of $P$.

At each of the endpoints $x_{k-1}$ and $x_k$, the function must have some well-defined value.

Step functions are also called piecewise constant functions.