View source for Symmetric derivative

A generalization of the concept of derivative to the case of set functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916101.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916102.png" />-dimensional Euclidean space. The symmetric derivative at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916103.png" /> is the limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916104.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916105.png" /> is the closed ball with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916106.png" /> and radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916107.png" />, if this limit exists. The symmetric derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s0916109.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161010.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161011.png" /> of a real variable is defined as the limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161012.png" /></td> </tr></table>

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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161014.png" /> of a real variable has a symmetric derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161015.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161016.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161017.png" /></td> </tr></table>

if

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and one of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161019.png" />,

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if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161021.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161022.png" /> has an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161023.png" />-th order derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161024.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161025.png" />, then there is (in both cases) a symmetric derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161026.png" />, and it is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161027.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1937)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.D. James,   "Generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091610/s09161028.png" />th primitives"  ''Trans. Amer. Math. Soc.'' , '''76''' :  1  (1954)  pp. 149–176</TD></TR></table>



====Comments====
In [[#References|[1]]] instead of derivative,  "derivate"  is used: symmetric derivate.