functions - How to add periodicity to condition?

Is it possible to set the following function f[a], where a can be any real number?

$$ f(\alpha) = \begin{cases} 1 & \text{if } 2\pi n < \alpha < \pi (2n+1) \\ 2 & \text{if } \pi (2n+1) \le \alpha \le 2\pi (n+1) \end{cases}, \ n \in \mathbb{Z}\ $$

The issue here is n which I want to be substituted consecutively by an integer in order to produce propriate continuous conditions. I guess, it is not the case where Assumptions should be used as my attempt underneath doesn't work.

f[a_] := \[Piecewise] {
   {1, 2 \[Pi]n < a && a < \[Pi] (2 n + 1)},
   {2, \[Pi] (2 n + 1) <= a && a <= 2 \[Pi] (n + 1)}
  }, Assumptions -> n \[Element] Integers

Thanks.

Answer

Here's one way to make your function periodic:

f[t_] := Which[0 <= t < Pi, 1, Pi <= t < 2 Pi, 2, t < 0, f[t + 2 Pi],  t > 2 Pi, f[t - 2 Pi]]

For example:

Plot[f[t], {t, -10, 10}]

This method works well for any function you care to use (not just square waves). For instance, f[t] can be linear in one half and quadratic in the second half:

f[t_] := Which[0 <= t < Pi, t, Pi <= t < 2 Pi, t^2, t < 0, f[t + 2 Pi], t > 2 Pi, f[t - 2 Pi]];
Plot[f[t], {t, -10, 10}]