On the Modulus of Continuity of Mappings Between Euclidean Spaces
- Autores: Dieudonné Agbor, Jan Boman
- Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 112, Nº 1, 2013, págs. 147-160
- Idioma: inglés
- DOI: 10.7146/math.scand.a-15238
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Resumen
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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