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August 27th, 2009, 08:40 AM   #1
Joined: Aug 2009

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On the Tietze extension theorem in the metric case.

If f is a uniformly continuous function defined on a closed subset A of a metric space with values in [1,2], its extension given by the formula:
F(x)= inf ( f(a)d(x,a) : a in A) / d(x,A)
is uniformly continuous as well: this is proven in a paper of Mandelkern 'on the uniform continuity of Tietze's extensions.
I am wondering what happens if one replaces "uniformly continuous" by Lipschitzean (allowing a different Lipschitz-constant for the extension).
I am tempted to believe it's wrong, but cannot find any couterexample...
Many thanks!

PS under the hypotheses of continuity only, this is well known; in many textbooks this formula is used often to provide an explicit extension; in the case of Lipschitzean functions, there are other ways to extend it to a Lipshcitzean function with the same constant, which are more natural. So this question, in this sense, is not "natural", yet correctly settled...
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