Density and Approximate Continuity

Abstract

The concepts of density and approximate continuity play a critical role in analysis of functions of several variables, with the important result being that for any set E ⊂ Rⁿ,

$$ \mathop{{\lim }}\limits_{{r \to 0}} \,\frac{{m*(B(x,r) \cap E)}}{{mB(x,r)}} = 1 $$

for almost all x ∈ E, m and m* now denoting Lebesgue measure and Lebesgue outer measure, respectively, in Rⁿ. The result is also true if we replace Lebesgue measure by Hausdorff measure (see Chapter 8). In this chapter, we prove the above result (Theorem 2.2.1) in the setting of R¹. We also show the connection between approximate continuity and integrability. Sierpinski’s theorem extends the concept of approximate continuity to nonmeasurable functions. We end this chapter with a discussion of the Darboux property and the density topology.