TITLE: The continuity of Injectivity radius
QUESTION [16 upvotes]: Dear all,
when reading a book of M. Berger, I learned that the injectivity radius Inj(x) on a compact Riemannian manifold depends continuously on the point x.
When the manifold is complete and non-compact, Inj may not be continuous. 
For example, Inj(x) decreases to zero when x moves to the most curved point on a paraboloid. However, it could be infinity at that point.
My question is, can we prove the continuity of Inj on a non-compact manifold under some conditions? 
(I think that the weakest condition is to assume the finiteness of Inj.)
ps. I must admit that I don't know how to prove the continuity of Inj even on a compact manifold. I think that the argument should involve the stability of ODEs (the geodesic equation and Jacobi equation). If one of you have a reference about this, could you please tell me? thanks a lot!

REPLY [12 votes]: The compactness is irrelevant; i.e if it is true for compact manifolds then the same is true for complete ones. (The same proof as in comact case works, but it is easier to do this way.)
If $R<\mathrm{InjRad}_p$ then one can construct a smooth metric on a sphere with an isometric copy of $B_R(p)$ inside.
If there is a sequence of points $x_n\to x$  such that
$$\lim\ \mathrm{InjRad}_{x_n}< \mathrm{InjRad}_x,$$
apply above consruction for $R$ slightly smaller than $\mathrm{InjRad}_x$.
You get a compact manifold with non-continuous InjRad.
If there is a sequence of points $x_n\to x$ such that 
$$\lim\ \mathrm{InjRad}_{x_n}> \mathrm{InjRad}_x$$ then apply above construction for $p=x_n$ for large enough $n$ and $R>\mathrm{InjRad}_x$.
That leads to a contradiction again.