  This paper is concerned with the study of the fractional finite sums theory.
We present the classes of functions for which it is possible to characterize
the constant related to the derivative of fractional sums (denominated by
essence of a function). Examples of the essences of functions are presented,
including the Bernoulli numbers, which are obtained as essences of the
functions $f(x)=x^a$, $a \in \mathbb{N}^*$. Based on the new results, we can
expand the comprehension of the Euler-Maclaurin summation formula for a real
number instead of only considering natural numbers. Moreover, we utilize the
concept of the essence of a function to propose a new regularization method for
divergent series.
