Lipschitz Continuity

A stronger requirement for continuity corresponds to some maximum rate of change of a function.

Definition

Consider two Metric Spaces $(X, d_{X})$ and $(Y, d_{Y})$ . A function $f : X \to Y$ is $L$ -Lipschitz continuous if

d_{y} (f (x_{1}), f (x_{2})) \leq L \cdot d_{x} (x_{1}, x_{2}), \forall x_{1}, x_{2} \in X .

Then, $L$ is a Lipschitz constant for $f$ .

More restrictively, for $X$ and $Y$ as Normed Vector Spaces or a Banach Spaces, we can rewrite this with the induced measures given by the norms as

| | f (x_{1}) - f (x_{2}) | |_{Y} \leq L \cdot | | x_{1} - x_{2} | |_{X}, \forall x_{1}, x_{2} \in X

Clearly, it is possible for this to hold for multiple Lipschitz constants, but we are often interested in the smallest Lipschitz constant.

If $X = Y = R$ , and $f$ is differentiable, then

\frac{d f}{d x} \leq L

is an equivalent definition. There are more general definitions based on derivatives for other spaces as well.