Analytic functions are both infinitely differentiable and have a Taylor series which converges in some neighbourhood of every point.

I think it's possible to argue that the functions of physics should not only be sufficiently differentiable, but analytic. The reason for thinking this – well, the reason I think it – is that I think these approximation techniques correspond to the measurements we can make: you can measure the position of something, say, and then by measuring the change in position over time you can measure velocity, and by measuring changes in that you can measure acceleration and so on.

Well, it would be nice if, based on the measurements we make, we can predict the system forwards1. And that's exactly what analytic functions give you.

  1. Classically. 

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