Smoothness via directional smoothness and Marchaud's theorem in Banach spaces

Classical Marchaud's theorem (1927) asserts that if f is a bounded function on [a, b], k is an element of N, and the (k + 1)th modulus of smoothness w(k+1) (f; t) is so small that eta(t) = integral(t)(0) omega(k+1)(f;s)/s(k+1) ds < +infinity for t > 0, then f is an element of C-k ((a, b)) and f((k)) is uniformly continuous with modulus C eta for some c > 0 (i.e. in our terminology f is C-k,C-c eta-smooth). Using a known version of the converse of Taylor theorem we easily deduce Marchaud's theorem for functions on certain open connected subsets of Banach spaces from the classical one-dimensional version. In the case of a bounded subset of R-n our result is more general than that of H. Johnen and K. Scherer (1973), which was proved by quite a different method. We also prove that if a locally bounded mapping between Banach spaces is C-k,C-w-smooth on every line, then it is C-k,C-w-smooth for some c > 0. (C) 2014 Elsevier Inc. All rights reserved.