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find all functions ?help

gantznil Fri 13 Mar 03:46:49 2020 No.11465049 View Reply Original Report

Quoted By: >>11465053 >>11465089 >>11465093

find all the functions such that they fulfill these conditions

Anonymous

Anonymous Fri 13 Mar 2020 03:48:43 No.11465053 Report

Quoted By:

>>11465049
Yes, I’m ready to meet some Asians in Seattle

Anonymous

Anonymous Fri 13 Mar 2020 04:06:54 No.11465089 Report

Quoted By: >>11465306

>>11465049
First property: https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
So f(x) = kx, now:
kab = f(ab) = f(a)f(b) = kakb = k2ab
Thus k = 1 or k = 0. The only two functions are either the indentity or constant equal to 0.

Anonymous

Anonymous Fri 13 Mar 2020 04:08:41 No.11465093 Report

Quoted By:

>>11465049
F(x)=x satisfies both your conditions

Anonymous

Anonymous Fri 13 Mar 2020 06:09:27 No.11465306 Report

Quoted By:

>>11465089
>So f(x) = kx
This only follows for functions from rationals to rationals, not reals to reals.
If you think about the additive structure of reals as a Q-vector space, you find a bunch of f's satisfying the first equation that aren't f(x)=kx

Anonymous

Anonymous ## Mod Fri 13 Mar 2020 11:54:36 No.11465713 Report

Quoted By:

This thread was moved to >>>/wsr/788044

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