I hear you, and I admit I'm drawing a fuzzy line (is the conventional approach “rigorous”).

Here are two “test functions”-

- we learned much about impulse responses, and sometimes considered responses to dipoles, etc. However, if I read the Wikipedia article correctly (it’s not great…), the theory implies that a distribution (in the technical sense) has derivatives of any order. I’m not sure I really knew that I could count on that. A rigorous treatment would have given me that assurance.

- if I understand correctly, the concept of introducing an impulse to a system that has an identity impulse response, which implies an inner product of delta with itself, is not well-defined. Again, I’m not sure if we covered that concept. (Admittedly, it’s been a long time.)