Afunction f : (a, b) → r satisfies a h¨older condition of order α if α > 0, and for some constant h and all u, x ∈ (a, b) we have |f(u) − f(x)| ≤ h|u − x| α the function is said to be α-h¨older, with α-h¨older constant h. (the terms "lipschitz function of order α" and "α-lipschitz function" are sometimes used with the same meaning.)
(a) prove that an α-h¨older function defined on (a, b) is uniformly continuous and infer that it extends uniquely to a continuous function defined on [a, b]. is the extended function α-h¨older?
(b) what does α-h¨older continuity mean when α = 1?
(c) prove that α-h¨older continuity when α > 1 implies that f is constant.