fTmp[t_ /; 0 <= t <= 2] := [Piecewise] {
   {1, t <= 1},
   {0, t <= 2}
  }
f[t_] := fTmp[Mod[t, 2]]

Plot[f[t], {t, 0, 5}]

NDSolve[{a'[t] == f[t]*a[t], a[0] == 1}, a, {t, 0, 5}]
... NDSolve::nlnum: The function value {2.71828 fTmp[2.18952]} is not a list of numbers with dimensions {1} at {t,a[t],NDSolve`s$49306[t]} = {2.18952,2.71828,1}.