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Approximately continuous functions are intimately connected to [[Lebesgue point]]s. For a function <math>f \in L^1(E)</math>, a point <math>x_0</math> is a Lebesgue point if it is a point of Lebesgue density 1 for <math>E</math> and satisfies
:<math>\lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x_0))} \int_{E\cap B_r (x_0)} |f(x)-f(x_0)|\, dx = 0</math>
where <math>\lambda</math> denotes the [[Lebesgue measure]] and <math>B_r(x_0)</math> represents the ball of radius <math>r</math> centered at <math>x_0</math>. Every Lebesgue point of a function is necessarily a point of approximate continuity.<ref>{{cite book |last=Thomson |first=B.S. |title=Real functions |publisher=Springer |year=1985 |isbn= |pages=}}</ref> The converse relationship holds under additional constraints: when <math>f</math> is [[essentially bounded]], its points of approximate continuity coincide with its Lebesgue points.<ref>{{cite book |last=Munroe |first=M.E. |title=Introduction to measure and integration |publisher=[[Addison-Wesley]] |year=1953 |isbn= |pages=}}</ref>
== See also ==
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