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Stronger than Lipschitz continuity on metric space

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Let $(M, d)$ be a metric space (which can be compact and/or connected if needed).

Is there a natural function class (of $M\rightarrow \mathbb{R}$-functions) that is strictly included in the Lipschitz continuous functions on $(M, d)$?

On the real line, the polynomials are among the most basic continuous functions, but those are not necessarily available in metric spaces.


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