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!

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!