Suppose that $f\colon [0,1]\to [0,1]$ is a continuous, surjective and nondecreasing function, for example the Cantor function. Let $X$ be a metric space (not necessarily a length space) and let $L$ be a length functional on $X$, satisfying the usual requirements (listed below). If $\gamma\colon [0,1]\to X$ is continuous, must $L(\gamma\circ f)=L(\gamma)$?
The requirements on $L$ are: (1) $L(\gamma|_{[a,c]})=L(\gamma|_{[a,b]})+L(\gamma|_{[b,c]})$, (2) $t\mapsto L(\gamma|_{[0,t]})$ is continuous, (3) $L$ is compatible with the topology of $X$, and (4) $L(\gamma)$ is invariant under strictly increasing changes of parameter (see page 27 of the book by Burago/Burago/Ivanov).
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