Is there a complete metric space $X$ with the following property?
For any pair of points $p,q\in X$ there is unique minimizing geodesic $[pq]_X$ that connects $p$ to $q$, but the map $(p,q)\mapsto [pq]_X$ is not continous.
Comments
For sure such space cannot be compact (or proper).
If we fix one end $p$, then there is a classical example --- the wheel; it is a unit circle with continuum of spikes from to the center $p$ to each point on the circle.
An example of noncomplete space with this property can be constructed by starting with a long circle and applying the following construction countably many times: Given a geodesic space $X$ construct a space $X'$ where to each pair of points $p,q\in X$ such that $|p-q|_X>1$ we add a unit segment with ends in $p$ and $q$.
