November 5th, 2017, 10:08 PM  #1 
Senior Member Joined: Nov 2011 Posts: 224 Thanks: 2  Dirchlet function
Are you have, please, a picture or web site with a picture that show the following Dirchlet function (I know that it is possible to draw this function exactly, but I want a close look on it): If x is irrational than y = x If x is rational y = 0. I think this function will be a Linear function of x = y despite if x is rational it is zero, and more think: I think that it can be proven by R.A. Can somebody, please, answer me... Thank you 
November 5th, 2017, 10:54 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,091 Thanks: 1087 
consider $f(\sqrt{2}x)$ if $f$ was linear then $f(\sqrt{2}x)=\sqrt{2}f(x),~\forall x$ let $x=\sqrt{2}$ $f(\sqrt{2}\sqrt{2})=f(2) = 0 \neq \sqrt{2}f(\sqrt{2})=2$ so $f$ is not linear 
November 5th, 2017, 11:00 PM  #3 
Senior Member Joined: Nov 2011 Posts: 224 Thanks: 2 
I don't understand. Please elaborate your message!! Thank you!!! You are welcome...!!! 
November 5th, 2017, 11:16 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,091 Thanks: 1087 
a linear function $f(x)$ must be such that i) $f(a x) = a f(x),~\forall a \in \mathbb{C}$ and ii) $f(x+y) = f(x)+f(y),~x,y \in \mathbb{C}$ so if your $f$ was linear $f(\sqrt{2}x) = \sqrt{2}f(x)$ and $f(\sqrt{2}\sqrt{2})= \sqrt{2}f(\sqrt{2})$ but $f(\sqrt{2}\sqrt{2})=f(2) = 0$ $\sqrt{2}f(\sqrt{2}) = \sqrt{2}\cdot \sqrt{2} = 2$ $0 \neq 2$ so $f$ is not linear 
November 5th, 2017, 11:27 PM  #5 
Senior Member Joined: Nov 2011 Posts: 224 Thanks: 2 
So There is noway to exemplified that function? a picture of it? It is not drawable. Or there is a way to draw a look of the function despite it is not drawable. Thank you for answering my question 
November 6th, 2017, 02:07 AM  #6  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,854 Thanks: 1078 Math Focus: Elementary mathematics and beyond  Quote:
 

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