# Jump function

One of the three components in the Lebesgue decomposition of a function of bounded variation depending on one real variable, rediscovered and extended to functions of several variables by De Giorgi and his school (see [AFP]). According to Lebesgue, if $I\subset\mathbb R$ is an interval, a right-continuous function of bounded variation $f: I\to\mathbb R$ can be decomposed in a canonical way into three functions $f_a+f_j+f_c$. The function $f_j$ is the jump part of $f$ (or jump function of $f$, using the terminology of Lebesgue [Le]) and it is defined by $$\label{e:jump} f_j (x)= \sum_{y\leq x}\, f (y^+) - f(y^-)\, .$$ Therefore its distributional derivative is the atomic part of the distributional derivative of $f$. The jump part can also be characterized as the (right-continuous) function $g$ with smallest variation such that $f-g$ is continuous (cp. with Remark 1 at page 163 of [Le]). Observe therefore that the total variation of $f$ is the sum of the total variation of $f_j$ and the total variation of $f-f_j$.
The term jump function is used also for those functions of bounded variation $f$ such that $f=f_j$, i.e. so that their distributional derivative is a purely atomic measure. See also Atom.