Let $M$ be a compact Riemannian manifold and let $SM$ be its sphere bundle,
$$SM = \{(x,\xi) \in TM : \|\xi\| = 1\}.$$
There is a well-defined function $\ell : SM \rightarrow [0,\infty)$ defined by mapping $(x,\xi)$ to the length of the unique maximally extended geodesic $\gamma$ with $\gamma(0) = x$ and $\gamma'(0) = \xi$. Is $\ell$ continuous?
If $\ell$ is continuous, then compactness immediately implies that the lengths of maximally extended geodesics on $M$ are bounded away from 0. If it turns out that $\ell$ is not continuous, is there a different way to prove this statement?
If I understand your definitions, the function $\ell$ isn't continuous. Consider a flat torus, $\mathbb{R}^2 / \mathbb{Z}^2$. The geodesic through $(0, 0)$ can be infinitely long if its angle is an irrational multiple of $\pi$, and will fluctuate wildly on the rational multiples of $\pi$.