2
$\begingroup$

I have a homework question, to prove that if $f(x)$ is continuous at $x_0 = 0$ and $f(0) = 0$ then $g(x)$ is continuous at $x_0=0$, where $g(x)=f(x)D(x)$ and $D(x)$ is the Dirichlet Function.

I am having trouble proving this, could anyone please help me out?

Thanks a-lot

  • 1
    I'm not sure "the Dirichlet function" has one standard definition - it would help to include the definition in your question.2011-11-25

1 Answers 1

2

You need that $f(0)=0$ otherwise you could take $f(x)=1$ which is continuous at $x_0=0$ but you have that $f(x)\cdot D(x)=D(x)$ which is not continuous at $0$.

If you have it however it is easy to see that is correct by using the fact that $|D(x)|\leq1$ everywhere. Therefore you have that $f(x)D(x)$ has to be $0$ at $x_0=0$ (I let you write out the details).

Edit: As you are asking for more details, I will give you another hint but the question is very trivial so I fear to completely solve it...

For $g(x)$ to be continuous at $0$ it is sufficient to show that

$\lim_{x \rightarrow 0}g(x)=g \Big(\lim_{x \rightarrow 0}x \Big)=g(0)=f(0)D(0)=0.$

Therefore you can use that

$\lim_{x \rightarrow 0}|g(x)|=\lim_{x \rightarrow 0}|f(x)D(x)|\leq \cdots \leq 0.$

Then your theorem follows, now you have to fill out the dots.

  • 0
    I am wondering what this question asking about and what is the word continous mean in this question?2011-11-25