delta less-than epsilon EndLayout"/>
and the Claim is proved.
A less restrict version of the Fundamental Theorem of Calculus is stated next. A proof of it follows as in [108, Theorem 9.6].
Theorem 1.75 (Fundamental Theorem of Calculus): Suppose is a continuous function such that there exists the derivative , for nearly everywhere on i.e. except for a countable subset of . Then, and
Now, we present a class of functions
Definition 1.76: A function
If we denote by
In
The next two versions of the Fundamental Theorem of Calculus for Henstock vector integrals, as described in Definition 1.41, are borrowed from [70, Theorems 1 and 2]. We use the term almost everywhere in the sense of the Lebesgue measure
Theorem 1.77: If and are both differentiable and is such that for almost every , then and , that is,
Theorem 1.78: If is differentiable and is bounded, then , and there exists the derivative for almost every , that is,
Corollary 1.79: Suppose is differentiable and nonconstant on any nondegenerate subinterval of and is bounded and such that . Then, almost everywhere in .
From Corollary 1.50, we know that if
Theorem 1.80: If and , then we have .
The next result is borrowed from [70, Theorem 5]. We reproduce its proof here.
Theorem 1.81: Suppose and is such that almost everywhere on . Then, and , that
