On the Modulus of Continuity of Mappings Between Euclidean Spaces - Agbor, Dieudonné; Boman, Jan (MATHEMATICA SCANDINAVICA, 2013)
Let $f$ be a function from $R^p$ to $R^q$ and let $Λ$ be a finite set of pairs $(θ, η) \in R^p × R^q$. Assume that the real-valued function $\langleη, f(x)\rangle$ is Lipschitz continuous in the direction $θ$ for every $(θ, η) \in Λ$. Necessary and sufficient conditions on $Λ$ 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 $\le Cε |{\log ε}|$.