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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?

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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).

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No, it means that $f(S) = A$, where $A$ is a finite set : $f$ take only a finite number of values

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  • $\begingroup$ Is this equivalent to the definition that a simple function is a linear combination of characteristic functions? $\endgroup$ – астон вілла олоф мэллбэрг Apr 7 '16 at 0:18
  • $\begingroup$ 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 $f = \sum_{k=1}^n a_k \chi_{f^{-1}(\{a_k\})}$ $\endgroup$ – Tryss Apr 7 '16 at 0:22
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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

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