Let $f$ be locally integrable, then for $x_0\in\mathbb{R}$ we have $$\lim\limits_{R\to 0}\frac{1}{|B_R(x_0)|}\int\limits_{B_R(x_0)}|f(x)-f(x_0)|dx=0.$$ The point $x_0$ is called Lebesgue point of $f$. Show that:
at $x_0$ has $f$ a jump discontinuity $\Rightarrow$ $x_0$ is not a Lebesgue point
This means that $\lim\limits_{x\uparrow x_0} f(x)\neq \lim\limits_{x\downarrow x_0} f(x)$. If $R$ is small than $B_R(x_0)$ contains either $\lim\limits_{x\uparrow x_0} f(x)$ or $\lim\limits_{x\downarrow x_0} f(x)$. So intuitively it is clear, but I do not know how to write it down.