MHB Prove that dirichlet function is periodic

[esuahcdss12]

Hey,

I need to prove that dirichlet function is a periodic function,
but i got no idea how to start solving the question.
could anyone help please?

 

[S.G. Janssens]

Do you mean the characteristic function of the rationals,
\[
f(x) =
\begin{cases}
1& (x \in \mathbb{Q})\\
0& (x \not\in \mathbb{Q})
\end{cases}
\]
?

 

[esuahcdss12]

Do you mean the characteristic function of the rationals,
\[
f(x) =
\begin{cases}
1& (x \in \mathbb{Q})\\
0& (x \not\in \mathbb{Q})
\end{cases}
\]
?

yes

 

[S.G. Janssens]

Can you prove that if $x$ is rational, then $x + 1$ is rational?
Can you prove that if $x$ is irrational, then $x + 1$ is irrational?

 

[HallsofIvy]

That would be sufficient to show that f is "periodic with period 1". Is there a "fundamental" (smallest positive) period?

 

[Greg]

What about a rational constant, say $c$, in the place of $1$?

 

[S.G. Janssens]

What about a rational constant, say $c$, in the place of $1$?

That would have been my next question...

That would be sufficient to show that f is "periodic with period 1". Is there a "fundamental" (smallest positive) period?

...and that the question after that (Nod)

I hope the OP still comes back. This function provides a nice exercise. Once the above is done, he could try to go on by proving (without looking it up somewhere) that this function is nowhere continuous, hence not Riemann integrable over any compact interval.

On the other hand, he could also investigate what a continuous function looks like when it is periodic with arbitrarily small periods.