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In this paper we study integer multiplicity rectifiable currents carried bythe subgradient (subdifferential) graphs of semi-convex functions on a$n$-dimensional convex domain, and show a weak continuity theorem with respectto pointwise convergence for such currents. As an application, the $k$-Hessianmeasures are calculated by a different method in terms of currents.

Currents carried by the subgradient graphs of semi-convex functions and applications to Hessian measure