Analytic extensions of differentiable functions defined in closed sets

Abstract

A continuous extension the author has not seen in the literature may be given as follows; we assume for simplicity that A is bounded.Let h(r) (räO) be a continuous and monotone increasing function such that A(0)=0, and if x and y are any two points of A whose distance apart is rxv, then \f(x) -f(y) \úh(rxv).For any points x of E and y of A, set B(x, y)=f(y)-h(rxi); then if x is in .4,B(x,y)^f(x).The continuous extension of f(x) is,F(x), which at each point x of £ equals the maximum of H(x, y) as y varies over A. 63