View source for Jump function

{{MSC|26A45|28A15}}

[[Category:Analysis]]

{{TEX|done}}

One of the three components in the [[Lebesgue decomposition|Lebesgue decomposition]] of a [[Function of bounded variation|function of bounded variation]] depending on one real variable, rediscovered and extended to functions of several variables by De Giorgi and his school (see {{Cite|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 {{Cite|Le}}) and it is defined by
\begin{equation}\label{e:jump}
f_j (x)= \sum_{y\leq x}\, f (y^+) - f(y^-)\, .
\end{equation}
Therefore its [[Generalized derivative|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 {{Cite|Le}}). Observe therefore that the [[Variation of a function|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]].

====References====
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|valign="top"|{{Ref|AFP}}||      L. Ambrosio, N.  Fusco, D.   Pallara, "Functions of bounded   variations   and  free  discontinuity   problems". Oxford Mathematical   Monographs. The     Clarendon Press,   Oxford University Press, New  York,  2000.       {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|DS}}||      N. Dunford, J.T. Schwartz,  "Linear operators. General theory",     '''1''', Interscience (1958)  {{MR|0117523}} {{ZBL|0635.47001}}
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|valign="top"|{{Ref|Le}}||  H. Lebesgue,   "Leçons sur l'intégration et la récherche des fonctions  primitives", Gauthier-Villars  (1928).
|-
|valign="top"|{{Ref|Na}}||  I.P. Natanson,   "Theorie der Funktionen einer reellen Veränderlichen" ,  H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)
|-
|valign="top"|{{Ref|Sa}}|| S. Saks,   "Theory of the integral" , Hafner  (1952) {{MR|0167578}} {{ZBL|63.0183.05}}
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|valign="top"|{{Ref|Sz}}|| B. Szökefalvi-Nagy,   "Introduction to real functions and orthogonal expansions" , Oxford Univ. Press  (1965)
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