Difference between revisions of "Primitive function"
From Encyclopedia of Mathematics
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| − | ''anti-derivative, of a finite function | + | {{TEX|done}} |
| + | ''anti-derivative, of a finite function $f$'' | ||
| − | A function | + | 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 | + | 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
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