  It is hereby established that, in Euclidean spaces of finite dimension,
bounded self-contracted curves have finite length. This extends the main result
of Daniilidis, Ley, and Sabourau (J. Math. Pures Appl. 2010) concerning
continuous planar self-contracted curves to any dimension, and dispenses
entirely with the continuity requirement. The proof borrows heavily from a
geometric idea of Manselli and Pucci (Geom. Dedicata 1991) employed for the
study of regular enough curves, and can be seen as a nonsmooth adaptation of
the latter, albeit a nontrivial one. Applications to continuous and discrete
dynamical systems are discussed: continuous self-contracted curves appear as
generalized solutions of nonsmooth convex foliation systems, recovering a
hidden regularity after reparameterization, as consequence of our main result.
In the discrete case, proximal sequences (obtained through implicit
discretization of a gradient system) give rise to polygonal self-contracted
curves. This yields a straightforward proof for the convergence of the exact
proximal algorithm, under any choice of parameters.
