The proofs here provided by Stephen M and Jack Lee 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!