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''anti-derivative, of a finite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074560/p0745601.png" />''
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''anti-derivative, of a finite function $f$''
  
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]].
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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====
 
====References====
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====Comments====
 
====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]].
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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]].
 
See also (the editorial comments to) [[Derivative|Derivative]].

Latest revision as of 20:24, 17 October 2014

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 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) 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, for bounded $F'$ — by the Lebesgue integral, and for any $F'$ — by the Denjoy integral in the narrow (or wide) sense and the Perron integral.

References

[1] L.D. Kudryavtsev, "A course in mathematical analysis" , 1 , Moscow (1981) (In Russian)
[2] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)


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.

See also (the editorial comments to) Derivative.

References

[a1] R.P Boas jr., "A primer of real functions" , Math. Assoc. Amer. (1981)
[a2] T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)
How to Cite This Entry:
Primitive function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_function&oldid=18563
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article