Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with no-zero finite length $L$. Then, does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and satisfies $lth\_t(g) = tL ~\forall ~t \in [0,1]$, where $lth\_t(g)$ represent the length of the function $g$ upto $x - coodinate = t$?


Any views would be really appreciated.



does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and ..?
This is a standard result that every rectifiable curve in a metric space admits an arc length parametrization. The proof can be found in many sources. For examples Theorem 3.2 in
[these](http://www.pitt.edu/~hajlasz/OriginalPublications/Hajlasz-SobolevSpaces-ContempMath-338-2003-173-218.pdf) notes. Then the arc-length parametrization is defined on $[0,L]$ and a linear change of variables leads us to $g$ defined on $[0,1]$.


