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 "https://mathoverflow.net/questions/53381/the-continuity-of-injectivity-radius", but the discussion there focuses on compact or complete manifolds.