And often we have functions which are only defined in an open neighborhood of zero, yet we still call them functions. That the set theoretic definition of "fun

And often we have functions which are only defined in an open neighborhood of zero, yet we still call them functions. That the set theoretic definition of "function" as a functional relation between sets is rarely useful isn't really so surprising that you need to emphasize that you use a more reasonable notion of function all the time. And the integral transform notation is frequently changed, but only ever locally and in different ways by different authors. Pick up two books on "Fourier transforms for engineers" and I promise you that you will find different notations for the same thing and probably even different notations within the same book. And none of them will be good.