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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., I need an upper bound on the minimum number of $\epsilon$-radius (open) norm balls needed to cover the set $R=\{f(x) | x \in D\}$. We may assume that $R$ is compact and if necessary also assume that $D$ is compact. I am completely knew to this kind of mathematics. I would like to thus know:

  • What is known about this number?
  • 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?

Edit: In case there is no general answer, I thought I will narrow down the question. I am actually interested in a rather specific case. My $f$ is the (unique) solution of an ODE. Thus, $D=[0,\infty)$ and $f$ is in $C^1$.

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  • $\begingroup$ What do you know about $f$? $\endgroup$
    – Qiaochu Yuan
    Commented Jan 13, 2015 at 7:09
  • $\begingroup$ I have added a few edits and more details about $f$. Thanks! $\endgroup$
    – Ankur
    Commented Jan 13, 2015 at 7:18

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