Proof that the Heaviside function is an element of $L_1^{loc}(\Bbb R)$

Question

Let be

$H(x) = \begin{cases} 1 & , \,\, x \ge 0 \\ 0 & , \,\, x < 0 \end{cases}$

and

$f(x) = \begin{cases} \log(|x|) & , \,\, x \ne 0 \\ 2016 & , \,\, x = 0 \end{cases}$.

My question:

I know that $H$ and $f$ are elements of $L_1^{loc}(\Bbb R)$, but knows someone a proof of that?

I don' t have any idea how one could show this.

What I know:

$H(x)$ is the heaviside function and its distributional derivation is the $\delta$ distribution.