# derichlet function question..

• Dec 17th 2008, 09:50 AM
transgalactic
derichlet function question..
derihlet function return 1 if the input is rational.
and 0 if the input is irrational.

prove that Dirichlet function is periodic

and prove that every rational number is its period
??
• Dec 17th 2008, 09:57 AM
Isomorphism
Quote:

Originally Posted by transgalactic
derihlet function return 1 if the input is rational.
and 0 if the input is irrational.

prove that Dirichlet function is periodic

and prove that every rational number is its period
??

I think all that needs is the following observation:

For a fixed rational r,
1) Every rational can be written as sum of another rational and r, uniquely.
2) Every irrational can be written as sum of another irrational and r, uniquely.

f(x+r) = f(x) for all x, where f is the Dirichlet function.
• Dec 17th 2008, 10:25 AM
transgalactic
f(x+r) = f(x)

how this expression proves the its periodic?
• Dec 17th 2008, 10:37 AM
Isomorphism
Quote:

Originally Posted by transgalactic
f(x+r) = f(x)

how this expression proves the its periodic?

That is the definition of periodicity I know... What is yours?
• Dec 17th 2008, 10:52 AM
transgalactic
thanks you are correct

how to prove the second part that
each rational number is its period?

i think i can use f(x+r) = f(x)and say that
for every rational number "r"

i get same number

is it ok?
• Dec 17th 2008, 09:37 PM
Isomorphism
Quote:

i think i can use f(x+r) = f(x)and say that
for every rational number "r"
Exactly!

For every r, f(x+r) = f(x) and thus r is the period of the function.

Thus every rational is the period of the function.