Rectangular Functions

Why

We represent rectangles by functions.

Definition

A rectangular function corresponds to a characterstic function of an interval. It represents a rectangle whose width is the interval and whose height is one.

Notation

Let $A$ be a non-empty set and $B \subset A$. Recall that we denote the characteristic function of $B$ by $\chi _{B}$.

Now suppose that $A \subset \R $. If we embed $\set{0, 1} = 2 \in \N $ in $\R $ by associating $0$ to $0_{\R }$ and $1$ to $1_{\R }$ then every characteristic function is identified with a function from $\R $ to $\R $.

In particular, notice that if $B$ is an interval and $\alpha $ is a real number then $\alpha \chi _{B}$ is a rectangle with height $\alpha $.