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 November 5th, 2017, 10:08 PM #1 Senior Member   Joined: Nov 2011 Posts: 197 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: 1,977 Thanks: 1026 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: 197 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: 1,977 Thanks: 1026 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: 197 Thanks: 2 So There is noway to exemplified that function? a picture of it? It is not draw-able. Or there is a way to draw a look of the function despite it is not draw-able. Thank you for answering my question
November 6th, 2017, 02:07 AM   #6
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 Originally Posted by shaharhada Are you have, please, a picture or web site with a picture that show the following Dirchlet function...
See here.

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