let $M$ a complete and compact riemannian manifold and let $\Omega$ a bounded open set in $M$. I would like to know why the function $i_{M}: M\to [0,\infty)$ is continuous when $M$ is compact, here $i_{M}$ is the radius of injectivity of $M$.
My question arise when the author in the papper which i'am reading asserts that
$i_{\bar{\Omega}}=inf \{i_{M}(x), x \in \bar{\Omega}\}>0$ and $i_{\bar{\Omega}} \leq min\{i_{M}(x_{0}),i_{M}(y_{0})\}$, where $x_{0},y_{0} \in \Omega$.