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.