Piecewise linear approximation of smooth functions of two variables

Given a piecewise linear (PL) function p defined on an open subset of Rn, one may construct by elementary means a unique polyhedron with multiplicities D(p) in the cotangent bundle Rn×Rn⁎ representing the graph of the differential of p. Restricting to dimension 2, we show that any smooth function f(x,y) may be approximated by a sequence p1,p2,… of PL functions such that the areas of the D(pi) are locally dominated by the area of the graph of df times a universal constant.