This paper studies the regularity of the minimum time function, T (·), for a control system with a general closed target, taking the state equation in the form of a differential inclusion. Our first result is a sensitivity relation which guarantees the propagation of the proximal subdifferential of T along any optimal trajectory. Then, we obtain the local C 2 regularity of the minimum time function along optimal trajectories by using such a relation to exclude the presence of conjugate times.