# Prove that dirichlet function is periodic

• MHB
• esuahcdss12
In summary, the conversation is discussing the proof of the Dirichlet function being a periodic function, specifically the characteristic function of the rationals. The conversation touches on the proof for rational and irrational numbers, the possibility of a fundamental period, and the function's continuity and integrability.
esuahcdss12
Hey,

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

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

Krylov said:
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

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?

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

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

greg1313 said:
What about a rational constant, say $c$, in the place of $1$?

That would have been my next question...

HallsofIvy said:
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.

## What is the Dirichlet function?

The Dirichlet function, denoted as D(x), is a mathematical function that takes on the value of 1 when its input is a rational number and 0 when its input is an irrational number.

## What does it mean for a function to be periodic?

A periodic function is a function that repeats its values at regular intervals. This means that for a given input, the function will produce the same output after a certain period of time.

## How do you prove that the Dirichlet function is periodic?

To prove that the Dirichlet function is periodic, we need to show that there exists a period p such that D(x+p) = D(x) for all values of x. We can do this by considering the cases where x is rational and where x is irrational and showing that the function repeats its values for both cases.

## What is the period of the Dirichlet function?

The period of the Dirichlet function is p = 1. This means that the function repeats its values every 1 unit, or in other words, it is a periodic function with a period of 1.

## Why is it important to prove that the Dirichlet function is periodic?

Proving that the Dirichlet function is periodic is important because it helps us understand the behavior of this function and its properties. It also allows us to use this function in various mathematical applications, such as in Fourier series and harmonic analysis.

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