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*x*_{k}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*S*_{0}, whereas the density of a derived distribution element (the equivalent of the local derivative) is an object from the processual class*S*_{n}(where*n*is the rank of the derived distribution).

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