It's not that simple https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(diffe.

It's not that simple https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(diffe... So it's important to keep in mind the difference between integral and anti-derivative. And note that as soon you say "within an epsilon" you're already on the other side of "not a lot of info" vs "infinite info". So global, local doesn't matter, especially since eg an analytic function is completely determined by it's derivatives at a single point (good ol Taylor's theorem).