In this case, the function f is sometimes called a special uniformly continuous map.

First note that uniformly continuous for a function f is stated as follows:

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.

The function on is not uniformly continuous because f* fails to be microcontinuous at an infinite point .

Every uniformly continuous function between metric spaces is continuous.

Every continuous function on a compact set is uniformly continuous.

In particular, every function which is differentiable and has bounded derivative is uniformly continuous.

By the Heine-Cantor theorem it is uniformly continuous in that set.

The generator of a uniformly continuous semigroup is a bounded operator.

A continuous real function on the closed unit interval is uniformly continuous.