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August 27th, 2009, 08:40 AM  #1 
Newbie Joined: Aug 2009 Posts: 1 Thanks: 0  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 Lipschitzconstant 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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case, extension, metric, theorem, tietze 
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