I am trying to characterize the set of first-order definable functions $f: \Bbb R \to \Bbb R$ and see what properties they have.
It is immediately clear that these functions are not all analytic everywhere, since you have things like the absolute value function, the nearest integer function, the "give me the 10'th bit of the binary expansion of this real number" function, and so on. All of these above functions are instead piecewise-analytic functions.
Being piecewise-analytic is a fairly broad notion, but it still isn't everything: there are some important first-order definable functions such as the Weierstrass function, which are not piecewise-analytic. This is instead an infinite series of cosine functions, which are all analytic.
Similarly, the Cantor function is first-order definable, and expressed fairly easily as the limit of a sequence of piecewise-analytic functions (in particular piecewise-constant functions).
Both of these series/sequences are themselves first-order definable and converge pointwise. So far, every first-order definable function I have seen is of this form.
Is every first-order definable function of this form, i.e. the limit of some first-order definable sequence of analytic (or piecewise-analytic) functions that converges pointwise to it?
(To keep it simple I wrote the above in terms of total functions $\Bbb R \to \Bbb R$ but also interested in partial functions, e.g. $f(x) = 1/x$.)