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While according to the mathematical analysis there is a function f(x) on a continuous domain of a variable x, according to the objectual philosophy, it is a primary distribution f(x) (considered as continuous under the specific meaning of this paper) on a realizable support domain (discrete) {x}. The primary distribution has a singular value of the dependent attribute as its local element, assigned to a singular support value by means of a local relation. This local element is the equivalent of a function value in a point, according to the classic mathematics.
The primary distribution f(x)
can have (if it is uneven) some derived distributions of different
ranks. The local elements of these distributions are made-up
from a finite and linear variation (of a certain rank) of the
dependent attribute, assigned to a variation x
by means of a local relation, x
which is the same in terms of the amount, regardless of the
distribution rank. These local elements are the equivalent of the
relations X.3.2.2.3 and X.3.2.2.4, provided that
to
be however less but not under
.
The invariant density of the linear distribution on an element
of derived distribution is under these circumstances the equivalent
of the local derivative from the classic differential calculus.
Attention! This density is assigned to an interval
(which
can be referred to as an object through its internal reference xk
from the primary distribution f(x)). Therefore, according to
the objectual philosophy, the derivative of a function cannot exist
only on a singular value (the derivative equivalent in a point
from the classic differential calculus. If you have carefully read
chapter 4 where the processual classes of objects have been
presented, you could also find out in this way that a primary
distribution element (the equivalent of the function value in one
point) is an object belonging to the processual class S0,
whereas the density of a derived distribution element (the
equivalent of the local derivative) is an object from the processual
class Sn (where n is the rank of the
derived distribution).
Copyright © 2006-2011 Aurel Rusu. All rights reserved.
