EDB — 15F

[15F] Difficulty:*.Let \((X_ 1,d_ 1)\) and \((X_ 2,d_ 2)\) metric spaces, with \((X_ 2,d_ 2)\) complete. Let \(A⊂ X_ 1\) and \(f:A→ X_ 2\) be a uniformly continuous function. Show that there is a uniformly continuous function \(g:\overline A→ X_ 2\) extending \(f\); In addition, the extension \(g\) is unique.

Note that if \(𝜔\) is a continuity modulus for \(f\) then it is also a continuity modulus for \(g\). (We assume that \(𝜔\) is continuous, or, at least, that it is upper semicontinuous).