What are examples of discrete functions that are monotonically increasing/decreasing?

Question

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

Answer 1

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 $x \geq y$.

Answer 2

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 $\mathbb Z$.