I'm trying to find properties for functions that cover the following properties and wondering if they have a formal name to search more efficiently.
The function $f(x)$ cover the following requirements:
- $\operatorname{Image}{f(\Bbb R)}$ is countably infinite.
- $\forall a,b \in \Bbb R$ with $a<b$, $\operatorname{Image}((a,b))$ is finite.
- The discontinuities of $f(x)$ for $x \in (a,b)$ are finite.
An example of this functions is: $\lfloor x\rfloor$ (the integer part of $x$).
Such a function is a piecewise constant function on $\Bbb R$ with infinite image.
With only finitely many discontinuities in bounded intervals, the behaviour between successive discontinuities is necessarily constant, or else the image would not be countable. Hence every function with the indicated properties is piecewise constant, and having an infinite image was a requirement.
Conversely, since a piecewise defined function can admit only countably many pieces (each piece being either an interval or a boundary point of another piece), the image of any piecewise constant function must be countable (or finite, if your definition of countable excludes that).