[nicf]

You're welcome! There's actually one more point that I thought of after sending that reply, which since you mentioned it again I should maybe flag.

Totally apart from physics, it may seem intuitively plausible that if you have a function f and (a) all f's derivatives exist everywhere, and (b) f(0), f'(0), f''(0), etc. are all zero, then f must be the zero function. This is actually also not true! For a counterexample, you can look at this article on bump functions: https://en.wikipedia.org/wiki/Bump_function.