# Primitive function

*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=33749