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Let $X$ be a Banach space and $BC(X)$ the space of all bounded closed subsets in $X$. It can be shown that $(BC(X),d_H)$ is a complete metric space (see this page for a definition of $d_H$). If $f:X\to X$ bounded. Does this imply that $F:BC(X)\to BC(X)$ given by $F(A)=\overline{\{f(a):a\in A\}}$ is locally Lipschitz continuous?

And if the map $f: X\to X$ is locally Lipschitz countinuous, is it automaticaly bounded?

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    Does this hold for $f:\mathbb R\to\mathbb R$ given by $f(x)=\sqrt[3]{x}$, which has arbitrarily large derivative near $0$?2012-05-10

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