How to describe a piece-wise function in words?

Question

Suppose we want to describe a peice-wise function in words. My last post was down-voted because it was not clear.

Here is what I'm trying to desribe.

Function $F:D\to\mathbb{R}$, $D\subseteq \mathbb{R}$, $\bigcup_{i=1}^{\infty}A_n=D$ and

$$F(x)=\begin{cases} F_1(x) & x=A_1\\ F_2(x) & x=A_2\\ F_3(x) & x=A_3 \\ ... & ... \\ ... & ... \\ \text{Undefined} & \text{Everywhere Else} \end{cases} $$

Here is my description

Consider piece-wise function $F:D\to\mathbb{R}$ such that $D\subseteq {\mathbb{R}}$ is partitioned into chosen subsets $A_1,A_2,...A_n$ with $F_1(x)$ defined on $x\in A_1$, $F_2(x)$ is defined on $x\in A_2$ and so on.

Is this clear enough?

Here is the original question

Answer 1

1. If you want to emphasize that $F$ is piecewise, and just give a name to the piecewise regions, you can say:

   Let $F:D\rightarrow \mathbb{R}$ be defined piecewise on a given partition $D_1,\ldots, D_n$ of $D$.

2. If you want to emphasize your interest in a particular partition ("chosen subsets") where $F$ is defined piecewise, you can say:

   Suppose the domain $D$ is partitioned into particular nonoverlapping subsets $D_1,\ldots, D_n$, and that $F$ is a real-valued function that can be defined piecewise within each region.

3. If you want to emphasize or define what it means to be piecewise, you can say:

   Consider a real-valued function $F$ defined piecewise on a domain $D$ as follows: $D$ is partitioned into nonoverlapping subsets $D_1, \ldots, D_n$, and there is a function $F_i:D_i \rightarrow \mathbb{R}$ defined on each region. The overall function $F$ is defined by $F(x) = F_i(x)$ whenever $x\in D_i$, which uniquely specifies $F$ throughout all of $D$.