Approximate mean limit of a function (${\lim }_{r\downarrow 0}\frac{1}{|B_r(x)|}{\int }_{B_r(x)}|u(y)-a|dy=0$)

An approximate mean limit of a function is a value satisfying the displayed condition. Functions in the BV space have such finite limits outside an approximate discontinuity set up to negligible sets in the appropriate measure. Their approximate jump points instead have two distinct half-ball limits.

A scalar function in the BV space has an approximate jump at a point when its mean on each of two half-balls approaches a different finite value. The half-balls are determined by a unit normal ${\nu }_u$. Reversing the normal exchanges the two BV traces on a hypersurface. Their collection is the jump set of a BV function.