View source for Primitive function

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''anti-derivative, of a finite function $f$''

A function $F$ such that $F'(x)=f(x)$ everywhere in the domain of definition of $f$. This definition is the one most widely used, but others occur, in which the requirements on the existence of a finite $F'$ everywhere are weakened, as are those on the equation $F'=f$ 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 $f$ given on an interval is that $f$ is continuous; necessary conditions are that $f$ 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 $F$ from $F'$ for continuous $F'$ is solved by the [[Riemann integral|Riemann integral]], for bounded $F'$ — by the [[Lebesgue integral|Lebesgue integral]], and for any $F'$ — 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 $f$ on an interval $I$ says that if $f(a)$ and $f(b)$ are two values of $f$, $a,b\in I$, then $f$ assumes any value between $f(a)$ and $f(b)$ at some point between $a$ and $b$. 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>

{{SourceattributionEOM|Primitive function}}
[[Category:Mathematics]]