On Extending Peano Derivatives

Let E be a closed set with inf E = a and sup E = b, and k be a positive integer. Let f : E Rbe such that the k-th Peano derivative of f relative to E, f _(k) (x, E), exists. It is proved under certain condition on the function f, that an extension F : a, b] Rof f exists such that the ordinary derivative of F of order k, F ^<k> (x) exists on a, b] and is continuous on a, b], and f _<> (x, E) = F ^<i> (x) on E, for i = 1, 2, &, k.