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It is known that nonconstant Hoelder functions $f:R \rightarrow R$ with power $>1$ non exist.

Are there functions $f: R\rightarrow R$ satisfying $ |f(x)-f(y)| \leq C|x-y|^p +D$ for all $x,y \in R$, where $C,D>0$, $p>1$?

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    Thanks. Yes, If there are.2012-08-01

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Any Lipschitz continuous (Hölder continuous with power $1$) function will do. Suppose $f:\mathbb R\to\mathbb R$ is such that $|f(x)-f(y)|\leq C|x-y|$ for all $x,y\in\mathbb R$. Then for any $p>1$ we have $|f(x)-f(y)|\leq C(|x-y|^p+1)=C|x-y|^p+C$ for any $x,y\in\mathbb R$, since if $|x-y|\leq 1$ then clearly $|x-y|< |x-y|^p+1$ while otherwise $|x-y|<|x-y|^p$.