ON THE MODULUS OF CONTINUITY OF MAPPINGS BETWEEN EUCLIDEAN SPACES

Let f be a function from R-P to R-q and let Lambda be a finite set of pairs (theta, eta) is an element of R-P x R-q. Assume that the real-valued function (eta, f(x)) is Lipschitz continuous in the direction theta for every (theta, eta) is an element of Lambda. Necessary and sufficient conditions on Lambda are given for this assumption to imply each of the following: (1) that f is Lipschitz continuous, and (2) that f is continuous with modulus of continuity <= C epsilon vertical bar log epsilon vertical bar.