I am a beginning graduate student. I have the following basic question I am very confused about:

Suppose C be a smooth geometrically irreducible curve over finite field F_q, q=p^m, p prime. Now take the ring A of functions on C regular away from a rational point say, \infty. And K is its function field and C_\infty be the completed algebraic closure of K wrt a valuation at \infty. now define a rigid analytic function say, f defined over C_\infty.

On the other hand suppose B= F_q[\theta], \theta indeterminate. and B_\infty defined same way as C_\infty. And then define the Tate algebra to be T_B = B_\infty[[T_1,•••••, T_l]], T_i are indeterminate different from \theta. Again define the rigid analytic function say, g over this Tate algebra T_B.

What is the difference between theory developed around the functions $f$ and $g$?