As remarked, any subadditive modulus of continuity is uniformly continuous: in fact, it admits itself as a modulus of continuity. Therefore, f∗ 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.
