Approximately continuous function: Difference between revisions

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== Definition ==

Let <math>E \subseteq \mathbb{R}^n</math> be a [[Lebesgue measurable set]], <math>f\colon E \to \mathbb{R}^k</math> be a [[measurable function]], and <math>x_0 \in E</math> be a point where the [[Lebesgue density]] of <math>E</math> is 1. The function <math>f</math> is said to be '''approximately continuous''' at <math>x_0</math> if and only if the [[approximate limit]] of <math>f</math> at <math>x_0</math> exists and equals <math>f(x_0)</math>.<ref>{{cite book |last=Federer |first=H. |title=Geometric measure theory |publisher=Springer-Verlag |series=Die Grundlehren der mathematischen Wissenschaften |volume=153 |___location=New York |year=1969 |isbn= |pages=}}</ref>

== Properties ==