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Hi All,

Suppose f1, f2, and g are arbitrary functions of real variables.

Let

f(x) =

f1(x) if x is rational,

f2(x) if x is irrational.

What software can be used to define f(x) such that we can plot y = f(g(x)) directly?

For example, consider the Thomae function:

f(x) =

1/q if x=p/q is rational, gcd(p,q)=1 and q>0,

0 if x is irrational.

How to define f as a function such that we can plot y = f(sin(x)) directly?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,369

Hi benice;

Looks like it is going to be tricky using the Dirichlet-Thomae function.

Just generating that function is a chore. The best solution I think is showed by J.M. over at the stackexchange. He generates lots of rationals using a farey sequence and then just plots them. See below first picture, it looks like yours.

How did you generate your points, did you use a farey sequence too?

Do you have a rationalize command in the language or grapher you are using? If you do then there is a way, see the second picture that is f(sin(x)), I think.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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bobbym wrote:

How did you generate your points, did you use a farey sequence too?

Hi,

I used the **GeoGebra** command 'Sequence' to generate those points:

Sequence[Sequence[(p/q,GCD[p,q]/q),p,1,q-1],q,2,300] .

bobbym wrote:

Do you have a rationalize command in the language or grapher you are using? If you do then there is a way

I have downloaded **Maxima** 5.28 (which has a rationalize command), but I have no idea how to do that.

Yesterday, I constructed an approximate method which can plot the graph of y = f(sin(x)) directly.

( See this page for some examples. )

Thank you for your help bobbym!

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,369

I have downloaded Maxima 5.28 (which has a rationalize command), but I have no idea how to do that.

The trick is to generate rationals. That is what Rationalize does.

1) Make a function called fubar in your language.

fubar[n_]:=Rationalize[Sin[x]] ( In mathematica )

2) Generate a sequence from 1/5000 to 1 with a step size of 1 /5000 using on fubar for each number.

3) Take the denominator of each of these numbers in the sequence and invert it. In other words 1 / denominator. That is how I generated the composition, f(g(x)).

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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