Lipschitz extensions and approximations

The classical Hahn-Banach theorem is based on a successive point-by-point procedure of extending bounded linear functionals. In the setting of a general metric domain, the conditions are less restrictive and the extension is only required to be Lipschitz with the same Lipschitz constant. In this case, the successive procedure can be replaced by a much simpler one which was done by McShane and Whitney in the 1930s. Using virtually the same construction, Czipszer and Gehér showed a similar extension property for pointwise Lipschitz functions. In the present paper, we relate this construction to another classical result obtained previously by Hausdorff and dealing with pointwise Lipschitz approximations of semi-continuous functions. Moreover, we furnish complementary extension-approximation results for locally Lipschitz functions which fit naturally in this framework.