Let $M$ be a differentiable manifold of dimension $n>2$ with a Riemannian metric $g=\sum\_{i,j=1}^ng\_{ij}dx\_idx\_j$ such that in some points on $M$ its coefficients $g\_{ij}$ are not differentiable (so $g\_{ij}$ are just continuous on $M$). Call $d\_g$ the metric induced on $M$ by $g$ (by the infimum of the lengths of the curves).


Is $(M,d\_g)$ a geodesic space (by "geodesic space" I mean that for every point $p\in M$ there is a neighborhood $U$ of $p$ such that for every couple of points $x,y\in U$ there exists a geodesic for $d\_g$ from $x$ to $y$) although $g$ has coefficients which are not differentiable at some points?


Clearly the problem is that in the geodesic equations $\displaystyle{\frac{d^2 x^a}{ds^2}+\Gamma^a\_{bc}\frac{d x^b}{ds}\frac{dx^c}{ds}=0}$ there are Christoffel symbols $\Gamma^a\_{bc}$ in which the derivatives of the $g\_{ij}$ appear. 



Geodesics for non differentiable riemannian metric
This is a length-metric, that is any two points $x$ and $y$ can be joined by a path with length arbitrary close to the distance from $x$ to $y$.
Further, your metric space is locally bi-Lipschitz to the Euclidean space,
in particular it is locally compact.


If your space is complete then by [Hopf–Rinow theorem](http://en.wikipedia.org/wiki/Hopf%E2%80%93Rinow_theorem) it is geodesic.


For sure your space is locally complete. Then the same proof imply that it is locally geodesic as you asked.


