The proofs provided here by [Stephen M](https://mathoverflow.net/a/335040) and [Jack Lee](https://mathoverflow.net/a/335826) now appear in my book about Riemannian optimization: [*An introduction to optimization on smooth manifolds*](http://www.nicolasboumal.net/book).  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!