# 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 k_{f}, 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).