Proposition 1

Every uniformly continuous mapping is continuous.

This is an immediate consequence of the definitions.

On the other hand, a continuous mapping of a uniform space X into a uniform space X' need not be uniformly continuous, \* as is shown by the example $x \to x^3$, a homeomorphism of $\mathbf{R}$ onto itself, which is not uniformly continuous with respect to the additive uniformity. \* (See § 4, no. 1, Theorem 2.)