Let $a<b$, and let $F:[a,b]\to\mathbb R$ have bounded variation. Regard $F$ as an element of $L^1((a,b),\mathcal B((a,b)),\mathcal L^1)$ when defining its [distributional derivative](/page/Distributional%20Derivative), and use the one-sided limits of the bounded-variation representative for jump sizes. Then there is a unique finite signed Borel measure $DF$ on $(a,b)$ such that, for every $\phi\in C_c^\infty((a,b))$, \begin{align*} \int_a^b F(x)\phi'(x)\,d\mathcal L^1(x)=-\int_{(a,b)}\phi\,dDF. \end{align*} The measure $DF$ has a unique decomposition \begin{align*} DF=g\,\mathcal L^1+D^jF+D^cF, \end{align*} where $g\in L^1((a,b),\mathcal B((a,b)),\mathcal L^1)$, the measure $D^jF$ is purely atomic, and the measure $D^cF$ is singular with respect to $\mathcal L^1|_{(a,b)}$ and has no atoms. More explicitly, \begin{align*} D^jF=\sum_{x\in J_F}\bigl(F(x+)-F(x-)\bigr)\delta_x, \end{align*} where $J_F\subset(a,b)$ is the at most [countable set](/page/Countable%20Set) of jump points and $F(x-),F(x+)$ denote the one-sided limits. The three measures $g\,\mathcal L^1$, $D^jF$, and $D^cF$ are called respectively the absolutely continuous part, the jump part, and the singular continuous part of the distributional derivative measure of $F$.