Lebesgue point

Summary

Let $f$ be a locally integrable function on $\R^n$. A Lebesgue point of $f$ is a point $x$ where the average of $\abs{f - f(x)}$ over small balls around $x$ tends to $0$. At such a point, $f$ is well represented by its value $f(x)$ in an averaged sense. By the Lebesgue differentiation theorem, almost every point is a Lebesgue point. Points of continuity are Lebesgue points, but Lebesgue points need not be points of continuity.