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November 5th, 2017, 11:08 PM   #1
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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
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November 5th, 2017, 11:54 PM   #2
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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
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November 6th, 2017, 12:00 AM   #3
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I don't understand. Please elaborate your message!!
Thank you!!!
You are welcome...!!!
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November 6th, 2017, 12:16 AM   #4
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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
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November 6th, 2017, 12:27 AM   #5
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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
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November 6th, 2017, 03:07 AM   #6
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Quote:
Originally Posted by shaharhada View Post
Are you have, please, a picture or web site with a picture that show the following Dirchlet function...
See here.
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