Definition

Let $f\in L^{1}(\mathbb{R})$. We say that $x \in\mathbb{R}$ is a Lebesgue point for $f$ if and only if $$\lim_{ r \to 0 } \frac{1}{2r}\int\limits _{(x-r,x+r)}(f-f(x) )\, dm =0$$ ## Remark $f$ continuous at $x$ $\implies$ $x$ is a Lebesgue point for $f$. Also, intuitively, $x$ is a Lebesgue point for $f$ if $f$ does not oscillate too much near $x$.