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?
On
No, it means that $f(S) = A$, where $A$ is a finite set : $f$ take only a finite number of values
On
The range of a function is the subset of the co-domain, in your case a subset of the real numbers, for which your function f has an "output value". If the range is finite, it simply means that that set of outputs is a finite set. Since the real numbers form an ordered set, any non-empty and finite subset will have a largest and smallest element, so it also follows that your function f is both bounded above and below
More than just bounded: it means what it says — the function has only finitely many values. In general it won't be continuous (it's continuous iff it's constant on each connected component of its domain).