For some reason, I need to justify use of the "function" $f(x) = 0$ if $x < 0$ and $f(x) = \infty$ if $x > 0$. Is there a theory which allows to use such a function?
Specifically, I looked to Sato hyperfunctions. However, all examples of Sato hyperfunctions I know (I read in the first two chapters of Urs Graf's "Introduction to Hyperfunctions and Their Integral Transforms", and skimmed several other books) only have isolated singular supports. Yet, as far as I know, I cannot find the proof of the thesis that singular supports of Sato hyperfunctions are isolated points.
I tried to construct an example myself, but it seems for me to be impossible. This is because, to do this, we need a homomorphic function with non-isolated singularity on the real axis, but only examples I can come up with are branching points, which are only have finite(?) gap.
I am not an expert in this area, so I should miss something very trivial. But I would appreciate any comment or suggestion regarding this.
