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.