 Solution

If a regular $C^3$ extension existed at either finite endpoint, reparametrization by arc length would extend the Frenet data. Because $\kappa \ge \epsilon$, the denominator in the torsion formula stays away from zero; compactness of a short closed parameter interval would then bound both $\kappa$ and $|\tau |$. This contradicts the stated limsup at that endpoint. Applying the argument at both ends proves nonextendibility.