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Can you please give me some example of Lipschitz contraction that is easy to visualize? Do I understand it right that in R the K=1 (Lipschitz constant) describe only the constant functions? Is it possible to have K<1 on R?

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I think you might be mixing up Lipchitz and Holder continuity, the former being:

$$|f(x)-f(y)|\leq K|x-y|$$

and the latter being:

$$|f(x)-f(y)|\leq K|x-y|^\alpha.$$

For Lipchitz continuity, you can have any non-negative $K$. If $K=0$ your function is constant. Otherwise let $f(x)=Kx$, so that $|f(x)-f(y)|=|K||x-y|$, implying $|K|$ can be anything.

For Holder continuity, if $\alpha>1$, then the function is constant (hint: write down it's derivative).

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