Potpourri, 11

Let (M, d(x, y)) be a metric space. Thus M is a nonempty set and d(x, y) is a nonnegative real-valued function defined for x, y ∈ M such that d(x, y) = 0 if and only if x = y, d(y, x) = d(x, y) for all x, y ∈ M , and d(x, z) ≤ d(x, y) + d(y, z) (1) for all x, y, z ∈ M. If d(x, z) ≤ max(d(x, y), d(y, z)) (2) for all x, y, z ∈ M , then we say that d(x, y) defines an ultrametric on M. A real-valued function f (x) on M is said to be Lipschitz of order α for some positive real number α if there is a nonnegative real number C such that |f (x) − f (y)| ≤ C d(x, y) (3) for all x, y ∈ M , which is equivalent to f (x) ≤ f (y) + C d(x, y) (4) for all x, y ∈ M. Notice that this holds with C = 0 if and only if f is constant on M. If p ∈ M and f (x) = d(x, p), then f is Lipschitz of order 1 with C = 1 by the triangle inequality. More generally, if A is a nonempty subset of M , then dist(x, A) = inf{d(x, y) : y ∈ A} (5) is Lipschitz of order 1 with C = 1. Every Lipschitz function on the real line of order α > 1 is constant. One way to see this is to observe that such a function is differentiable at every point with derivative equal to 0. More generally, every Lipschitz function on R n of order strictly larger than 1 is constant.