PlanetMath: Dirichlet's function

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Dirichlet's function

(Definition)

Dirichlet's function $f:\mathbb{R}\to\mathbb{R}$ is defined as

$\begin{displaymath} f\left(x\right) = \left\{ \begin{array}{ll} \frac{1}{q} & \t... ...m{if } x \textrm{ is an irrational number.} \end{array}\right. \end{displaymath}$

This function has the property that it is continuous at every irrational number and discontinuous at every rational one.

Another function that often goes by the same name is the function

$\begin{displaymath} f\left(x\right) = \left\{ \begin{array}{ll} 1 & \textrm{if }... ...m{if } x \textrm{ is an irrational number.} \end{array}\right. \end{displaymath}$

This nowhere-continuous function has the surprising analytic expression

$\displaystyle f(x) = \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2 n} (m! \pi x).$

This is often given as the (amazing!) example of a sequence of everywhere-continuous functions whose limit function is nowhere continuous.

"Dirichlet's function" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Cross-references: continuous, limit function, sequence, expression, rational, discontinuous, irrational number, continuous at, property, function
There are 5 references to this entry.

This is version 6 of Dirichlet's function, born on 2002-12-01, modified 2008-03-25.
Object id is 3639, canonical name is DirichletsFunction.
Accessed 8758 times total.

Classification:

AMS MSC:

26A15 (Real functions :: Functions of one variable :: Continuity and related questions )

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

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How to integrate the Dirichlet function by vatai on 2004-10-08 15:38:25

hi all!

as i recall the dirichlet fn.

(f(x)=1/q if x is rational and x=p/q (in the lowes term) and f(x)=0 if x is irational.)

is integrable, and (i dont know on what interval) is its integral =0 (i think the improprius integral, on all x in R)

But how to proove it's integrable, and that the integral is zero?!?

thanx,
emil

[ reply | up ]

Re: How to integrate the Dirichlet function by mathforever on 2004-10-08 16:15:34
- Re: How to integrate the Dirichlet function by vatai on 2004-10-08 19:10:29
  - Re: How to integrate the Dirichlet function by mathforever on 2004-10-09 17:30:30
    - Re: How to integrate the Dirichlet function by vatai on 2004-10-09 18:31:57
      - Re: How to integrate the Dirichlet function by mathforever on 2004-10-10 10:08:08
        
        Re: How to integrate the Dirichlet function by rspuzio on 2004-10-10 13:33:19
        
        Re: How to integrate the Dirichlet function by rspuzio on 2004-10-10 14:12:29
        
        Re: How to integrate the Dirichlet function by mathforever on 2004-10-10 16:06:53
        
        Re: How to integrate the Dirichlet function by rspuzio on 2004-10-10 17:35:56

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