D[FractionalPart[x], x] == x - Sum[KroneckerDelta[k, 0] DiracDelta[x - k], {k, -Infinity, Infinity}]

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "x"], RowBox[List["FractionalPart", "[", "x", "]"]]]], "\[Equal]", RowBox[List["x", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["k", ",", "0"]], "]"]], RowBox[List["DiracDelta", "[", RowBox[List["x", "-", "k"]], "]"]]]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <mi> frac </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> x </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> k </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> x </ci> </bvar> <apply> <ci> FractionalPart </ci> <ci> x </ci> </apply> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["x_"]]], RowBox[List["FractionalPart", "[", "x_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["x", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["k", ",", "0"]], "]"]], " ", RowBox[List["DiracDelta", "[", RowBox[List["x", "-", "k"]], "]"]]]]]]]]]]]]

2001-10-29