Editing Modulus of continuity (section)

===Subadditive moduli, and extendibility===
The above property for uniformly continuous function on convex domains admits a sort of converse at least in the case of real-valued functions: that is, every special uniformly continuous real-valued function ''f'' : ''X'' → '''R''' defined on a metric space ''X'', which is a metric subspace of a normed space ''E'', admits extensions over ''E'' that preserves any subadditive modulus ω of ''f''. The least and the greatest of such extensions are respectively:

:<math>\begin{align}
f_*(x) &:=\sup_{y\in X}\left\{f(y)-\omega(|x-y|)\right\}, \\
f^*(x) &:=\inf_{y\in X}\left\{f(y)+\omega(|x-y|)\right\}.
\end{align}</math>

As remarked, any subadditive modulus of continuity is uniformly continuous: in fact, it admits itself as a modulus of continuity. Therefore, ''f''<sub>∗</sub> and ''f*'' are respectively inferior and superior envelopes of ω-continuous families; hence still ω-continuous. Incidentally, by the [[Kuratowski embedding]] any metric space is isometric to a subset of a normed space. Hence, special uniformly continuous real-valued functions are essentially the restrictions of uniformly continuous functions on normed spaces. In particular, this construction provides a quick proof of the [[Tietze extension theorem]] on compact metric spaces. However, for mappings with values in more general Banach spaces than '''R''', the situation is quite more complicated; the first non-trivial result in this direction is the [[Kirszbraun theorem]].