View source for Primitive function

''anti-derivative, of a finite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745601.png" />''

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745602.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745603.png" /> everywhere in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745604.png" />. This definition is the one most widely used, but others occur, in which the requirements on the existence of a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745605.png" /> everywhere are weakened, as are those on the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745606.png" /> everywhere; a [[Generalized derivative|generalized derivative]] is sometimes used in the definition. Most of the theorems on primitive functions concern their existence, determination and uniqueness. A sufficient condition for the existence of a primitive function of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745607.png" /> given on an interval is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745608.png" /> is continuous; necessary conditions are that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745609.png" /> should belong to the first Baire class (cf. [[Baire classes|Baire classes]]) and that it has the Darboux property. Any two primitive functions of a function given on an interval differ by a constant. The task of finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456010.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456011.png" /> for continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456012.png" /> is solved by the [[Riemann integral|Riemann integral]], for bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456013.png" /> — by the [[Lebesgue integral|Lebesgue integral]], and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456014.png" /> — by the [[Denjoy integral|Denjoy integral]] in the narrow (or wide) sense and the [[Perron integral|Perron integral]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Kudryavtsev,   "A course in mathematical analysis" , '''1''' , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1''' , MIR  (1977)  (Translated from Russian)</TD></TR></table>



====Comments====
The Darboux property or intermediate-value property of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456015.png" /> on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456016.png" /> says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456018.png" /> are two values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456021.png" /> assumes any value between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456023.png" /> at some point between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p07456025.png" />. Continuous functions have the intermediate-value property (the intermediate-value theorem). The fact that the derivative of a one-time differentiable real-valued function on an interval has the intermediate-value property is sometimes referred to as Darboux's theorem. It is an immediate consequence of the [[Rolle theorem|Rolle theorem]].

See also (the editorial comments to) [[Derivative|Derivative]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P Boas jr.,   "A primer of real functions" , Math. Assoc. Amer.  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR></table>