I was thinking about this for a bit.
What are the finite functions that satisfy the following three criteria?
For instance, we have the boxcar function .
And then I can’t really think of anything else. It seems somewhat nontrivial coming up with such functions with larger support. I’ve seen an example somewhere, so it ought to be possible, but I can’t remember it.
There’s also the uninteresting example of partitioning the unit interval into a finite number of smaller intervals , whence the function is constructed. This allows arbitrarily large supports (as defined by the range) but also seems too much like a hack and isn’t quite what I’m looking for.
A “trivial” result is that if we assume that the support is an interval (open or closed doesn’t matter) and that the function is positive throughout said interval, then this interval must have length at most 1 (otherwise ) and therefore has to be the boxcar function or a translated version thereof.