Let's see... Reparametrize for arclength, and let the torsion . By continuity, over an interval .
If the Frenet-Serret frame is , then is also the position vector. We therefore have where is the curvature of . These last equations give or , where .
This DE is linear and thus solvable throughout I. Solve to get , for some constant vector . This means , and the parametrization gives , differentiating which gives . This implies over I, a contradiction as the curve lies on the sphere.