Originally Posted by

**Kramer** The Lebesgue Differentiation Theorem states:

If $\displaystyle f: \mathbf{R}^{n} \rightarrow C$ is a summable function then for almost every $\displaystyle x \in \mathbf{R}^{n}$ the following limit converges to $\displaystyle 0$:

$\displaystyle lim_{r \rightarrow 0}\text{ } (\frac{1}{m(B(x,r)})\int_{B(x,r)}|f(y) - f(x)|dy = 0$

Where $\displaystyle m$ is the lebesgue measure and $\displaystyle B(x,r)$ is an open ball of radius $\displaystyle r$.

How does it follow that this Theorem is satisfied if $\displaystyle f$ is only assumed to be continuous?