TITLE: Is the injectivity radius (semi) continuous on a non-complete Riemannian manifold?
QUESTION [13 upvotes]: Let $\mathcal{M}$ be a Riemannian manifold, and let $\operatorname{inj} \mathrel\colon \mathcal{M} \to (0, \infty]$ be its injectivity radius function.
It is known that if $\mathcal M$ is connected and complete, then $\operatorname{inj}$ is a continuous function: see for example [Lee, Introduction to Riemannian Manifolds, 2018, Prop. 10.37].
What is known in the case where $\mathcal M$ is not complete? Is $\operatorname{inj}$ also continuous? If not, is there a known counter-example? Would $\operatorname{inj}$ still be semi-continuous?
This question is similar to the question "The continuity of Injectivity radius", but the discussion there focuses on compact or complete manifolds.

REPLY [5 votes]: The proofs provided here by Stephen M and Jack Lee now appear in my book about Riemannian optimization: An introduction to optimization on smooth manifolds.  See Section 10.8.
The answer is: yes, the injectivity radius is a continuous function even if the Riemannian manifold $\mathcal{M}$ is not complete.
Thank you both!