  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 Geh\'er 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.
