  We study in this note the stability and inclusion of the jump set of
minimizers of convex denoising functionals, such as the celebrated
"Rudin-Osher-Fatemi" functional, for scalar or vectorial signals. We show that
under mild regularity assumptions on the data fidelity term and the
regularizer, the jump set of the minimizer is essentially a subset of the
original jump set. Moreover, we give an estimate on the magnitude of jumps in
terms of the data. This extends old results, in particular of the first author
(with V. Caselles and M. Novaga) and of T. Valkonen, to much more general
cases. We also consider the case where the original datum has unbounded
variation, and define a notion of its jump set which, again, must contain the
jump set of the solution.

Analysis of PDEs;Classical Analysis and ODEs;Optimization and Control