[uuwp]

I don't think every continuous function can be approximated this way because we can make an infinitely complex, but continuous function that would have any n-th derivative also continuous. I'm thinking about those weird zeta-riemann-style functions. In order to approximate such a function we'd need a huge model that couldn't be computed or stored even by a universe-size perfect computer.

[DavidSJ]

It’s a theorem, so it’s been proven: https://en.wikipedia.org/wiki/Universal_approximation_theore...

Another caveat that I forgot in my previous comment is the domain has to be compact (closed and bounded). But if so, then it doesn’t really matter how weird your continuous function is, because compactness of the domain guarantees uniform continuity, i.e. your delta only depends on epsilon and not x in the epsilon-delta criterion of continuity. That allows you to partition the domain into patches of diameter delta, in which very simple functions are sufficient to approximate within epsilon.