  A classical result of variational analysis, known as Attouch theorem,
establishes the equivalence between epigraphical convergence of a sequence of
proper convex lower semicontinuous functions and graphical convergence of the
corresponding subdifferential maps up to a normalization condition which fixes
the integration constant. In this work, we show that in finite dimensions and
under a mild boundedness assumption, we can replace subdifferentials (sets of
vectors) by slopes (scalars, corresponding to the distance of the
subdifferentials to zero) and still obtain the same characterization: namely,
the epigraphical convergence of functions is equivalent to the epigraphical
convergence of their slopes. This surprising result goes in line with recent
developments on slope determination (Boulmezaoud, Cieutat, Daniilidis, 2018),
(P\'erez-Aros, Salas, Vilches, 2021) and slope sensitivity (Daniilidis,
Drusvyatskiy, 2023) for convex functions.
