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Topic: When standard differentiation & integration doesn't work
Replies: 8   Last Post: Sep 19, 2012 8:00 PM

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Porky Pig Jr

Posts: 1,535
Registered: 12/6/04
Re: When standard differentiation & integration doesn't work
Posted: Sep 19, 2012 8:00 PM
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On Tuesday, September 18, 2012 12:46:00 PM UTC-4, Victor Porton wrote:
> shoes6 wrote:
>

> > When standard calculus is used, one of the requirements for
>
> > differentiation is that a function be continuous.
>
> >
>
> > But consider the function f on the interval (0,1) defined thus:
>
> > f(x) = 1 if x is irrational
>
> > f(x) = 0 if x is rational
>
>
>
> You want to differentiate this function.
>
>
>
> I developed a theory for taking (generalized) limits for EVERY function.
>
> Thus we can easily define differentiation of EVERY function using my theory.
>
>
>
> Read:
>
> http://www.mathematics21.org/binaries/limit.pdf
>
> and before this consult:
>
> http://www.mathematics21.org/algebraic-general-topology.html
>
>
>
> Note that by EVERY function I mean every function from a topological vector
>
> space to a topological vector space, and similar cases.
>
>
>
> I don't know about such a generalization of integration, yet.
>
>
>
> --
>
> Victor Porton - http://portonvictor.org


I'm sure if you enhance you theory with Cantor-Finlayson set theory, such a generalization of integration will come just naturally.

PPJr.



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