Let $(X,d)$ be a metric space, and $x,y\in X$. In Probability Measures on Metric Spaces of Nonpositive Curvature, Sturm defines a geodesic joining $x$ and $y$ as some continuous path $\gamma :[a,b]\to X$ such that $\gamma(a)=x$, $\gamma(b)=y$ and property (i) holds, where
(i) $\forall r<s<t\in [a,b]$, $d(\gamma_r,\gamma_t) = d(\gamma_r,\gamma_s)+d(\gamma_s,\gamma_t)$.
Sturm states that (i) can be equivalently replaced with (ii), where
(ii) $\forall s<t\in [a,b]$, $L(\gamma_{\mid [s,t]}) = d(\gamma_s,\gamma_t)$.
This is not the standard definition of a geodesic joining $x$ and $y$. In Bridson's book and in Papadopoulos's book, a geodesic is an isometry, i.e., $\forall s,t\in [a,b]$, $d(\gamma_s,\gamma_t) = |s-t|$, which implies that $\gamma$ has unit speed.
My question: I wonder if Sturm's geodesic must have constant speed, i.e. there exists some $v\geq 0$ such that $\forall s,t\in [a,b]$, $d(\gamma_s,\gamma_t) = v|s-t|$.
Using property (i) we have a continuous function $f:[a,b]\mapsto [0,\infty)$ that verifies the functional equation $$\forall r<s<t\in [a,b], \quad f(r,t)=f(r,s)+f(s,t).$$ I tried to show the only solutions of this equation are of the form $(s,t)\mapsto v|s-t|$, without success.