the "If" part is trivial.
for the " only if" part: since h is periodic function with period 1, we only need to focus on the closed interval [0,1].
To prove h is positive constant, it is surffice to prove that h is nondecreasing in [0,1] which implies h is constant in [0,1].
this can be Proved by contradiction!
suppose there is two points in [0,1] such that .
then ,where is arbitrary positive integer. since f is increasing, we have
hold for any arbitrary positive integer .
since , let k approches infinity, then
tend to negative infinity, which is contradict to (*).
thus h is nondecreasing in [0,1]. combined with h(0)=h(1) gives that h is constant in [0,1], therefore constant in all real line.
since f is increasing, the constant is definitely positive, of course.