For an n-dim noncompact manifold M without boundary. Assume M contains a line $\gamma$. For every point $p \in M$, let $\widetilde{p\gamma(t)}$ be the geodesic from p to $\gamma(t)$.
Choose a subsequence $t_i \to \infty$, such that $\widetilde{p\gamma(t_i)}$ converge to a ray $\gamma_p$ from p to infty which is asymptotic to $\gamma$ .
Consider the Buseman function $\beta (\cdot)=\lim_{t\to \infty}(t-|\gamma(t)\cdot|)$. Then along the ray $\gamma_p$, $\beta$ grows with the speed 1. So we get $\beta$ is regular at every point p in M with $|\nabla\beta(p)|=1$ ?