What does it mean for simple functions to have finite range

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Solution 1

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).

Solution 2

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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Updated on April 07, 2020

Comments

  • Shamisen Expert
    Shamisen Expert over 3 years

    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?

  • Sarvesh Ravichandran Iyer
    Sarvesh Ravichandran Iyer over 7 years
    Is this equivalent to the definition that a simple function is a linear combination of characteristic functions?
  • Tryss
    Tryss over 7 years
    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 $f = \sum_{k=1}^n a_k \chi_{f^{-1}(\{a_k\})}$