I have here three step-wise functions, the first two in the image below, $f\, and \ g$, have four intervals, while their sum, $f+g=h$ has 8 intervals.
The first two are defined respectively on, $f: [-2,2]$ and $g: [-3,3]$ respectively.
I want to define the partitioning of these functions, which I call $\mathscr{P}$.
The partitioning of f, $\mathscr{P}_f$, and the partitioning of $g$, $\mathscr{P}_g$ are related to one another. Since the interval of $g$ is larger than the interval of $f$, then can one say that $\mathscr{P}_f\subset\mathscr{P}_g$?
Furthermore, $f+g=h$ which is imaged at the bottom in green (while f and g are in red and blue respectively). $h$ has 8 intervals, so its partitions are surely a subset of f and g. Can one then say that $\mathscr{P}_h\subset\mathscr{P}_f\subset\mathscr{P}_g$?
Thanks
