Let f be defined on an interval I. Suppose that there exists M > 0 and α > 0 such that for all x, y ∈ I. (Such a function is said to satisfy a Lipschitz condition of order α on I.) (a) Prove that f...

Let f be defined on an interval I. Suppose that there exists M > 0 and α > 0 such that

for all x, y ∈ I. (Such a function is said to satisfy a Lipschitz condition of order α on I.)

(a) Prove that f is uniformly continuous on I.

(b) If α > 1, prove that f is constant on I. (Hint: First show that f is differentiable on I.)

(c) Show by an example that if α = 1, then f is not necessarily differentiable on I.

(d) Prove that if g is differentiable on an interval I, and if g ′ is bounded on I, then g satisfies a Lipschitz condition of order 1 on I.