Take $f:[0,1]\to [0,1]^n$ a continuous tour through $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper bounds?

\begin{equation} \max_{|a-b|<\varepsilon} \|f(a)-f(b)\|. \end{equation}

Or if not a maximum, then the typical value for such $a,b$.

It seems that most research focuses on the opposite, more impressive direction. That is, for $\varepsilon>0$ characterizing how often $p,q\in [0,1]^n$ have their closest tour points $f(a)\approx p,f(b)\approx q$ such that $|a-b|<\varepsilon.$

For a k-th approximation to a Hilbert curve over $[0,1]^n$ is it true that for any $\varepsilon$-length interval roughly traverses not much more than a cube of ($\mathbb{R}^n$-)volume $\varepsilon$? This seems true from the construction....