Continuity of covariant derivative

Question

Suppose there exists a piecewise continously differentiable curve on a riemannian manifold. If at a point, the covariant derivative of the velocity vector field along the curve is continous, does it also imply that the curve is continously differentiable at that point?

Answer 1

If continuity is measured by one-sided limits, no: The path $\gamma(t) = (t, |t|)$ in the plane satisfies $D_{\gamma'}\gamma' = 0$ everywhere (in the sense of having extant, equal one-sided limits at each point), but $\gamma$ is not continuously differentiable at the origin.

The same is true of any non-smooth piecewise-geodesic path.