On the Modulus of Continuity of Mappings Between Euclidean Spaces
Keyword(s):
The Real
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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 $\le C\epsilon |{\log \epsilon}|$.
1975 ◽
Vol 78
(2)
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pp. 263-281
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1982 ◽
Vol 23
(2)
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pp. 137-149
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2001 ◽
Vol 192
(4)
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pp. 565-576
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2020 ◽
1986 ◽
Vol 01
(04)
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pp. 997-1007
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