# Primitive function

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anti-derivative, of a finite function

A function such that everywhere in the domain of definition of . This definition is the one most widely used, but others occur, in which the requirements on the existence of a finite everywhere are weakened, as are those on the equation 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 given on an interval is that is continuous; necessary conditions are that 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 from for continuous is solved by the Riemann integral, for bounded — by the Lebesgue integral, and for any — 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)