Let $f \in L^1(\mathbb{R};\mathbb{R})$ be a function of bounded variation,in the sense that \begin{eqnarray} \sup\limits_{\phi \in C_c^1(\mathbb{R}) \\ |\phi|_{L^{\infty}} \leq 1 }\int\limits_{\mathbb{R}} f\phi_x < \infty. \end{eqnarray}
Since $f\in L^1(\mathbb{R}),$ the distributional derivative of $f$ denoted by $f'$ is a signed radon measure. How to prove that $\int\limits_{\mathbb{R}} |f'|dx < \infty?$