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Covering number of the range of a function

I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. i.e … One, albeit weak, bound is simply the covering number of the ball in which $R$ lies. What is the best known result here? What body of literature has results like these? …