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.