1. ## Dirichlet function

I am trying to prove that f(x)=xD(x) is only continuous at x=0, however I don't know if I can prove it with assuming two different cases for continuity, in the neighbourhood of x=a when x is rational and irrational.. Not sure if it's possible to assume this, would appreciate if I can get feedback. Thank you.(In the picture I meant proving continuous , not convergence)

2. ## Re: Dirichlet function

Originally Posted by davidciprut
I am trying to prove that f(x)=xD(x) is only continuous at x=0, however I don't know if I can prove it with assuming two different cases for continuity, in the neighborhood of x=a when x is rational and irrational.. Not sure if it's possible to assume this, would appreciate if I can get feedback.
In every neighborhood there are both rational and irrational numbers. So of course you need cases.

3. ## Re: Dirichlet function

So my proof is correct?

5. ## Re: Dirichlet function

Originally Posted by MINOANMAN
Where is the problem? I know that D(x) is not continuous at any point but f(x)=xD(x) is continous at x=0.

6. ## Re: Dirichlet function

Originally Posted by davidciprut
Where is the problem? I know that D(x) is not continuous at any point but f(x)=xD(x) is continous at x=0.
You are correct. The function
is continuous at zero and nowhere else.

If . That shows it is continuous at zero.

If any neighborhood of x contains both rational and irrational numbers which means that f cannot be continuous at x.