Fourier series. If I'm not being slow, writing the functions as Fourier series of period T allows one to remove the first condition, as the zeroth Fourier coefficient being fixed. Then you want to navigate with the sum of the absolute squares being fixed. Basically this is paths on a sphere with one dimension taken out. There's a rotation that would do this, isn't there?
'''Edit''': The question as posed afresh involves a uniform norm bound also (at most 0.5 from the constant function 0.5, to put in one way).