So the solution to f(x^y) =f(x)*f(y) is that there does not exist a contionuos real function defined on reals>=0 such that f(x) is not 1 or 0 for all x.
Right?
Then the only option is to look for more interesting solution outside ( or rather, in between) real numbers.
If between any 2 real numbers there is a certain (e.g. infinite) amount of e.g. hyperreal numbers, than, for the purposes of this function, one can try to decide what function of hyperreals would satisfy this equation.
What is required is that such function has to have values 0 and 1 for all reals.
So for x=real , x>=0 f(x) = 1 or 0. We can choose 2 hyperreal values infinitely close to any real number x- e.g on both sides of it- so that one of them h1(x) < x would give value f(h1(x))=0, while other h2(x)>x would give value of function f(h2(x))=1.
Then we have a function f( h1(x), h2(x)) = 0 if arg=h1(x), 1 if arg=h2(x).)
Since h1(x) and h2(x) is infinitely close to x, in reals this function falls apart into 2 separate functions, or one that is not quite continuous.
In this case, f becomes a function only of hyperreals as it is 2 constant functions reals. We can use either discrete function which oscillates between 1 and 0 like
f(h(x)) = 1,0,1,0,1,0,1 for any 2 consequtive hyperreals in interval between 2 reals, or
we can form a one to one correspondence between all hyperreal interval between 2 neighbouring real numbers and Pi/2 and then f(h(x)) = sin (h(x)) where h1(x) = pi/2 and h2(x) = 0.
This correspondence between hyperreals and pi/2 is something I can not handle yet. Has it ever been shown what is the cardinality of hyperreals and what is the cardinality of pi/2?
Ivars
Right?
Then the only option is to look for more interesting solution outside ( or rather, in between) real numbers.
If between any 2 real numbers there is a certain (e.g. infinite) amount of e.g. hyperreal numbers, than, for the purposes of this function, one can try to decide what function of hyperreals would satisfy this equation.
What is required is that such function has to have values 0 and 1 for all reals.
So for x=real , x>=0 f(x) = 1 or 0. We can choose 2 hyperreal values infinitely close to any real number x- e.g on both sides of it- so that one of them h1(x) < x would give value f(h1(x))=0, while other h2(x)>x would give value of function f(h2(x))=1.
Then we have a function f( h1(x), h2(x)) = 0 if arg=h1(x), 1 if arg=h2(x).)
Since h1(x) and h2(x) is infinitely close to x, in reals this function falls apart into 2 separate functions, or one that is not quite continuous.
In this case, f becomes a function only of hyperreals as it is 2 constant functions reals. We can use either discrete function which oscillates between 1 and 0 like
f(h(x)) = 1,0,1,0,1,0,1 for any 2 consequtive hyperreals in interval between 2 reals, or
we can form a one to one correspondence between all hyperreal interval between 2 neighbouring real numbers and Pi/2 and then f(h(x)) = sin (h(x)) where h1(x) = pi/2 and h2(x) = 0.
This correspondence between hyperreals and pi/2 is something I can not handle yet. Has it ever been shown what is the cardinality of hyperreals and what is the cardinality of pi/2?
Ivars