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What are examples of discrete functions that are monotonically increasing/decreasing? Can monotonically increasing/decreasing be defined over discrete functions?

mallea
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2 Answers2

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Consider the function $f:\mathbb{Z} \mapsto \mathbb{Z}$ defined by $f(x)=x$. This function is monotonically increasing, since for every $x,y$ in the domain of $f$, $f(x) \geq f(y)$ whenever $x \geq y$.

Joe
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Yes they can.

Go look at your definition of what it means to be monotonic - all that's required is that some sense of "$\lt$" be defined on both the domain and range of your function.

Also, note that if you already have a monotonic function on a large, non-discrete set, and restrict its domain to a smaller, discrete subset, the restricted function is still monotonic. The gives you a huge set of examples by taking monotonic functions from $\mathbb R \rightarrow \mathbb R$ and restricting their domain to $\mathbb Z$.

JonathanZ
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  • (+1) Unfortunately, there's no agreed terminology. Some people say 'increasing' to mean $>$ and 'non-decreasing' for $\geq$; others use 'strictly increasing' and 'increasing' respectively. And I'm not sure whether 'monotonically increasing' is unambiguous either. – Joe Jan 13 '21 at 15:06
  • Yes, the term "monotonically increasing" is ambiguous, and that ambiguity is only cleared up when someone uses either of the forms "non-decreasing" or "strictly increasing". But the OP seemed mostly to be asking if one can define monotonicity for functions over discrete spaces, and for that question you only need the concept of "$\lt$". (Defining any of $\le$, $\gt$, or $\ge$ would also work just as well.) – JonathanZ Jan 13 '21 at 19:38
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    Good point! I will use non-decreasing and strictly increasing in future correspondence. – Joe Jan 13 '21 at 19:40
  • Good luck! I try to be precise, but usually end up reverting to sloppy and lazy. :--) – JonathanZ Jan 13 '21 at 23:13