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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.

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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 it is geodesic.

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

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