Best uniform approximation of semi-Lipschitz functions by extensions

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In this paper we consider the problem of best uniform approximation of a real valued semi-Lipschitz function \(F\) defined on an asymmetric metric space \((X,d),\) by the elements of the set \(\mathcal{E}_{d}(\left. F\right\vert _{Y})\) of all extensions of \(\left.F\right\vert _{Y}\) \((Y\subset X),\) preserving the smallest semi-Lipschitz constant. It is proved that this problem has always at least a solution, if \((X,d)\) is \((d,\overline{d})\)-sequentially compact, or of finite diameter.

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semi-Lipschitz functions; uniform approximation; extensions of semi-Lipschitz functions