# Diameters of the images of two balls under a function

Let $\Omega$ be an open and bounded subset of $\mathbb{R^n}$, and let $f:\Omega \to \mathbb{R}$ be a continuous function. I'm looking for some (preferably, minimal) conditions on $f$ under which, for some $M \ge 0$,

$\frac{diam(f(B(x,2r)))}{diam(f(B(x,r)))} \le M$

for all $B(x,2r) \subseteq \Omega$ when $diam(f(B(x,r))) > 0$.

Any help would be greatly appreciated.

Thanks.

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