# Metric Spaces, The Lipschitz Constant

## The Lipschitz Constant

The lipschitz constant (biography) is not a universal constant like E or π. Instead, the lipschitz constant is a property of a function f from one metric space into another. It is technically accurate to refer to this constant as kf, since it depends on f, although most textbooks simply use the letter k, where the function f is understood. As we shall see, there are some functions with no lipschitz constant. In this case we say k is undefined. Here is the formal definition.

Let f be a function from one metric space into another. The nonnegative real number k acts as a lipschitz constant for the function f if, for every x and y in the domain, the distance from f(x) to f(y) is no larger than k times the distance from x to y. We can write the relationship this way.

|f(x),f(y)| ≤ k × |x,y|

Technically, the lipschitz constant for f is the greatest lower bound of all k satisfying the above criteria. We know the condition fails when k is negative. The smallest possible value of k is 0. This can happen when f maps the entire domain onto one point. Since the values of k are bounded below, let's see why the greatest lower bound is also a lipschitz constant. Let k be the greatest lower bound, so that k+ε satisfies the above inequality, for any ε > 0. Now suppose we have found a pathological x and y such that |f(x),f(y)| exceeds k times the distance from x to y. In fact, let the distance from f(x) to f(y) be k+δ times the distance from x to y, where δ > 0. Set ε to half of δ, and find a contradiction. Thus the greatest lower bound is a valid lipschitz constant. If any k satisfies the above inequality for all of f, then f has a unique lipschitz constant, determined by taking the greatest lower bound of all such k.

In practice we rarely need to find the (minimal) lipschitz constant. It is often sufficient to demonstrate the existance of such a constant, whence any k will do. In other applications, any k less than 1 will suffice.

If f has a lipschitz constant k, (any k will do), then f is uniformly continuous. Select x in the domain and y in the range such that y = f(x). Restrict attention to an open ball about y, with radius ε. The open ball about x, with radius δ, where δ = ε/k, maps into the open ball about y. So far this looks like the definition of continuity, but nothing in this argument depends on x. It works for every x in the domain, hence δ is determined by ε, and the function is uniform.

As a corollary, a function that is not uniform, even a continuous function that is not uniform, cannot have a lipschitz constant. An example is the function 1/x on the open interval (0,1).