# What does it mean for simple functions to have finite range

In Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics By Dan Simovici, Chabane Djeraba, it says:

A simple function is a function $f: S \to \mathbb{R}$ that has finite range.

Can someone clarify what it means by "finite range"? Does it mean that $f$ is bounded below and above?

No, it means that $f(S) = A$, where $A$ is a finite set : $f$ take only a finite number of values
• Yes, it is. Every linear combination of characteristic functions has finite range, and every finite range function is a linear combination of characteristic functions. If the range is $\{a_1, \cdots, a_n\}$, then $f = \sum_{k=1}^n a_k \chi_{f^{-1}(\{a_k\})}$ – Tryss Apr 7 '16 at 0:22