Let $M$ be a smooth Riemannian manifold, $x$ be a point in $M$, and $\lambda:[0,1]\to M$ be a continuous path. Can we find a family of length-minimizing constant speed geodesics $\gamma_t:[0,1]\to M$ connecting $x$ and $\lambda(t)$, such that $t\mapsto\gamma_t(1/2)$ (i.e., the midpoint of the geodesic $\gamma_t$) is continuous on some interval $I\subset[0,1]$?
It appears that an affirmative answer to (a generalized version of) the above question is taken for granted in the penultimate paragraph in the proof of Theorem 9.2 in Villani's optimal transport book. Actually, it even asserted $t\mapsto\gamma_t(1/2)$ is continuous on the whole $[0,1]$, which is unfortunately wrong (consider the sphere and $\lambda$ being an geodesic arc passing through the antipodal of $x$).
It's unclear to me whether the question in the beginning can be proved or disproved in an elementary way (it's even unclear whether it is true). Any remedy to the proof of Theorem 9.2 aforementioned instead of the solution to this question is also appreciated.
