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Discontinuous Analysis |
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PP: 1-14 |
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doi:10.18576/jant/140101
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Author(s) |
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V. L. Porton,
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Abstract |
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| In this article, I investigate in detail (generalized) limit of an arbitrary (discontinuous) function, defined in terms of funcoids
(funcoids are briefly considered in this work, for people unfamiliar with them). Definition of generalized limit makes it obvious to
define such things as derivative of an arbitrary function, integral of an arbitrary function, sum of arbitrary series, etc. It is given a
definition of non-differentiable solution of a (partial) differential equation. It’s raised the question how do such solutions “look like”
starting a possible big future research program. It helps to calculate series, derivatives, integrals without first checking that they exist.
This theory allows you to check this once in the end of the calculation, instead of checking several times in the middle. The generalized
derivatives and integrals are linear operators. For example R b
a
f(x)dx −
R b
a
f(x)dx = 0 is defined and true for every function. This
has an advantage over (“competing” with my theory) distributions theory based analysis, that for example any two functions in my
analysis are multiplicible, while in distributions analysis you need to check a complex condition before multiplying two functions.
This is a straightforward, “no cost” advantage over traditional ways of non-smooth analysis. Moreover, in distributions analysis not
every function has a derivative, but in my analysis every function is differentiable. The advantages are further bettered by the fact
that I consider (generalized) limits of any values, not for a limited class of functions. The generalized solution of one simple example
differential equation is also considered. |
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