On the Modulus of Continuity of Mappings Between Euclidean Spaces

Dieudonné Agbor, Jan Boman

Abstract


Let $f$ be a function from $\mathbf{R}^p$ to $\mathbf{R}^q$ and let $\Lambda$ be a finite set of pairs $(\theta, \eta) \in \mathbf{R}^p \times \mathbf{R}^q$. Assume that the real-valued function $\langle\eta, f(x)\rangle$ is Lipschitz continuous in the direction $\theta$ for every $(\theta, \eta) \in \Lambda$. 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 $\le C\epsilon |{\log \epsilon}|$.

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DOI: http://dx.doi.org/10.7146/math.scand.a-15238

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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