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.