Be Rational

Today, enjoy a home-baked puzzle in real analysis--a somewhat old-fashioned subject that I can't help but love (much like automata theory).

We know that pi = 3.1415... is transcendental, i.e., it is not a zero of any nontrivial univariate polynomial with rational coefficients. Is it possible we could generalize this result?


Part I. Is there a continuous function f: R --> R, such that

a) f takes rational numbers to rational numbers,

b) f(x) is zero if and only if x = pi?


Part II. If you think no such f exists, then for which values (in place of pi) does such an f exist?
On the other hand, if f as in Part I does exist (I reveal nothing!), can it be made infinitely differentiable?


For those new to real analysis, 'weird' objects like the desired f above generally have to be created gradually, through an iterative process in which we take care of different requirements at different stages. We try to argue that 'in the limit', the object we converge to has the desired properties. (Does this sound like computability theory? The two should be studied together as they have strong kinships, notably the link between diagonalization/finite extension methods and the Baire Category Theorem.)

Passing to the limit can be subtler than it first appears; for example, a pointwise limit of continuous functions need not be continuous. Here is a pretty good fast online introduction to some central ideas in studying limits of functions.