The compactness is irrelevant; i.e if it is true for compact manifolds then the same is true for complete ones.

The fact that InjRad is lower semicontinuous is obvious;
i.e. if  $x_n\to x$ then 
$$\liminf\ \mathrm{InjRad}_{x_n}\ge \mathrm{InjRad}_x$$
I do not see a proof of upper semicountinuity,
BUT let me show that if it is true for compact manifolds then the same true for complete ones.

If $R<\mathrm{InjRad}_p$ then one can construct a smooth metric on a sphere with an isometric copy of $B_R(p)$ inside.

Now 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...

**NB!** I did not claim that InjRad is  upper semicontinuous, I only trust M. Berger.