# Primitive function

(Redirected from Anti-derivative)
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.