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For a complete riemannian manifold $ M $ with a point $ p_0\in M $, we have already kbow that the exponential map $ exp_{p_0}(v):\mathbb{S}^{m-1}\subset T_{p_0}M\rightarrow M $ is well defined. Meanwhile we can define $ \mu_{p_0}(v)=\sup\left\{s>0:d(exp_{p_0}(sv),p_0)=s\right\} $. Clearly we see that $ exp_{p_0}(\mu_{p_0}(v)v) $ is the cut point on the geodesic $ exp_{p_0}(tv) $ for all $ v\in \mathbb{S}^{m-1}\subset T_{p_0}M $. I want to show that the map $ \mu_{p_0}:\mathbb{S}^{m-1}\rightarrow \mathbb{R}_+ $ is continuous. I have tried to use some statements about conjugate points and Jacobi field, but I cannot complete the proof. Can anyone give me some hints about it ? References about such statement are also good.

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    $\begingroup$ Here is a paper which shows that the distance is not only continuous, but in fact uniformly Lipschitz for compact manifolds. However, I believe it's not possible to do much better than that for a generic metric, since cut loci can be quite irregular. jstor.org/stable/221962 $\endgroup$
    – Gabe K
    Jun 21, 2021 at 17:18
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    $\begingroup$ I should point out that $\mu$ need not be finite, and as such maps into $\mathbf{R} \cup \{ \infty \}$. $\endgroup$
    – Leo Moos
    Jun 23, 2021 at 16:29

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