# Is periodic but has no period ?

1. Sep 25, 2010

### lovemake1

1. The problem statement, all variables and given/known data

Today i came across this one question where i had no clue of how to proove.

it says f(x) = { 1, x rational
...................{ ---------------------is periodic but have no period. (Proove it/Show it)
..................{ 0, x irrational

ignore the dots and dashes., just for indentation.

2. Relevant equations

3. The attempt at a solution

I understand so far that 1/1 makes 1 rational, and 1/0 makes 0 irrational.
but how do we know if f(x) = 1 and f(x) = 0 is periodic?
arnt they just verticle line along x axis ? at y=1 and y=0

im very confused, im not even sure if i understand what the question is asking.

2. Sep 25, 2010

### Hurkyl

Staff Emeritus
This doesn't make sense,

nor this.

Could you try explaining what you mean in more detail?

(Just to check, do you know what f(3/7) and f(pi) are? My apologies if the question is too simple)

What is the definition of "f is periodic"? What is the definition of "f has no period"? (If your answer is "'f has a period' is false", then what is the definition of "f has a period"?)

3. Sep 25, 2010

### lovemake1

ok sorry for making the question so complicated.

This is how exactly it is worded.
"Show that function f(x) = 1 , x rational is periodic but has no period."
"Show that function f(x) = 0, x irational is periodic but has no period."

4. Sep 26, 2010

### Office_Shredder

Staff Emeritus
That doesn't make any sense as written. Probably what is intended is

Let f(x) be defined by
f(x)=1 if x is rational
f(x)=0 if x is irrational

Show that f(x) is periodic but has no period

So you're going to need to know two things
1) What does it mean for a function to be periodic?
2) What does it mean for a function to have a period?

5. Sep 26, 2010

### Hurkyl

Staff Emeritus
I would take this to be the statement of two separate problems.

In the first problem f is defined to be the constant function 1 with domain the rational numbers.
In the second problem, f is defined to be the constant function 0 with domain the irrational numbers.

But your original problem statement suggested something more like what Office Shredder wrote.

6. Sep 27, 2010

### HallsofIvy

There is an ambiguity here, perhaps intentional- a function, f, is periodic if there exist a number, P, such that f(x+ P)= f(x). "A" period of a periodic function is any P such that that is true. Of course, if f(x+ P)= f(x), then f(x+ 2P)= f((x+P)+ P)= f(x+P)= f(x) so any multiple of a period is also a period. The period of a periodic function is the smallest non-zero such P.

So a function can be periodic but "have no period" if and only if it has no smallest positive period. Any constant function has that property because every real number is a period and there is no smallest positive real number.

For this function, let x be any real number, d any rational number. Then
1) if x is rational, x+ d is also rational (the rationals are closed under addition) so f(x+ d)= 1 = f(x).

2) if x is irrational, x+ d is also irrational (If not, if x+ d= p, a rational number, then x= p- d= p+ (-d) which is impossible because the rationals are closed under addition) so f(x)= 0= f(x+ d)

Thus, this function is periodic with any rational number as "a" period. It "has no period" because there is no smallest positive rational number.

Last edited by a moderator: Sep 27, 2010
7. Sep 27, 2010

### lovemake1

Function is periodic but has no period?

1. The problem statement, all variables and given/known data

Show that the function f(x) = 1, x rational is periodic but has no period.
Show that the function f(x) = 0, x irrational is periodic but has no period.

2. Relevant equations

3. The attempt at a solution

I really have no clue how to approach this question,
im in trasitional phase from highschool math to university calculus math.

And this is one of thoes math problems i just zone out... and the question is worded very strangely for me. not understanding properly.

well i know that it is periodic because its a horizontal line along x axis.
and im not sure why it doesnt have a period ? maybe because its infinitely long?

can someone help me proove this ?
it would really help me.

8. Sep 27, 2010

### epenguin

Re: Function is periodic but has no period?

It does sound weird at first sight (like that phrase) yet it is almost common sense! What does it mean to say a function f(x) is periodic? For a start f(x) = ?

9. Sep 27, 2010

### eumyang

10. Sep 27, 2010

### cristo

Staff Emeritus
Re: Function is periodic but has no period?

11. Sep 27, 2010

### lovemake1

i understand that functions have period when (x+p) = f(x) = (x+p+p) because there are intervals that exist, hence f(x) equals original value at these points.

what do you mean by no smallest positive rational number?
could you clarify that part?

12. Sep 27, 2010

### Staff: Mentor

13. Sep 27, 2010

### Staff: Mentor

No, this isn't the definition. A function f is periodic with period p if f(x + p) = f(x) for all x.
Unlike, say, the positive integers, there is no smallest rational number. If you propose a rational number r as the smallest, I will counter with r/2. No matter which rational number you choose, I can come up with another one that is smaller.

14. Sep 27, 2010

### epenguin

"functions have period when (x+p) = f(x) = (x+p+p)"

You probably meant f(x + p) = f(x) = f(x + p + p) - what you wrote is not the definition of a periodic function.

You could just write f(x) = f(x + p) . It is actually true that, as you write, f(x) = f(x + p + p) - or more conventionally f(x) = f(x + 2p) - but that is not part of the definition of periodic it is something that follows from the definition of periodic. Think, if f(x) = f(x + p) for all x in the domain and for some p then you can argue in various ways, and I hope see, that f(x) also = f(x + 2p). Which also = what else? In general?

Now take the first part of your problem "Show that the function f(x) = 1, x rational is periodic". Now take any rational number, say 2, for x. What is f(x) in this case - i.e. what is f(2) according to that definition of f? And take another rational number, say 1/4 for p. What is f(x + p) for these values? If this makes the idea clear, write it down for any rational x and p.

This exercise is not about complicated calculations. It is not about knowing anything. It is about giving yourself permission to think (though in a logical mathematical way) and to express the thought using symbols.

15. Sep 28, 2010

### HallsofIvy

I suspect that the function involved is really "f(x)= 1 if x is rational, f(x)= 0 if x is irrational" and this is just one problem.