It is clear from the construction that this function has compact support.

The function 'f' is an example of a smooth function with compact support.

This has compact support in x, so is a compacton.

(notice this integral is actually over a finite region, since has compact support).

Let denote the space of locally constant functions on with compact support.

These are necessarily smooth functions of compact support in Ω.

This is just standard induction restricted to functions with compact support.

If f has compact support, then the last integral vanishes, and we have the desired result.

B has compact support and is an even function.

In most cases one demands of those functions to be piecewise continuous with compact support.