Show that Theorem 1.6.19 holds whenever $f$ is continuous.
The quantitative estimate needed is the following:
Theorem 1.6.20 (Hardy-Littlewood maximal inequality). Let $f$ : $\mathbf{R}^d \rightarrow \mathbf{C}$ be an absolutely integrable function, and let $\lambda>0$. Then
$$
m\left(\left\{x \in \mathbf{R}^d: \sup _{r>0} \frac{1}{m(B(x, r))} \int_{B(x, r)}|f(y)| d y \geq \lambda\right\}\right) \leq \frac{C_d}{\lambda} \int_{\mathbf{R}}|f(t)| d t
$$
for some constant $C_d>0$ depending only on $d$.