# Functions that vanish at integers

#### imAwinner

We know that the function f(x) = sin(2*pi*x) vanishes at all integers, are there other functions like that and what is the appropriate generalization to higher dimensions?

Cheers

#### Gib Z

Homework Helper
This is probably not what you are looking for, but unless you define a function like; f is a function that is equal to some constant y for non-integer values, and 0 for integer values; then you will find there is a very limited range of functions you can have.

You could also just put an integer K into the argument on the sine. And put any constant outside the sine.

#### ZioX

There are as many as you can draw. And more.

#### CRGreathouse

Homework Helper
There are as many as you can draw. And more.
i dont no... i can draw like a hundred of those... and id ont think there are more than that...

:tongue:

#### imAwinner

Maybe I wasn't specific, I'm looking for the formula of a function f(x,y) that would vanish at integers, could anyone help?

#### CRGreathouse

Homework Helper
We know that the function f(x) = sin(2*pi*x) vanishes at all integers, are there other functions like that and what is the appropriate generalization to higher dimensions?
For one-dimensional functions $f(x): \mathbb{R}\to\mathbb{R}$, there are
$$\beth_1^{\beth_1}=\beth_2=2^{2^{\aleph_0}}$$ ("lots")
of functions that vanish on the integers.

#### ZioX

Maybe I wasn't specific, I'm looking for the formula of a function f(x,y) that would vanish at integers, could anyone help?
Suppose we had two such real valued functions on the reals. What new function could you create such that its restriction on Z^2 is identically zero and nonzero on R^2/Z^2?

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#### HallsofIvy

Homework Helper
First of all note that you haven't required that the functions be continuous so you can pretty much pick any values you want for non-integer x and require f(n) be 0 for integer n. How about f(x)= 1 if x is not an integer, 0 if x is an integer?

Even requiring continuous, it is always possible to be whatever (constant) value you want between n and -n and and add a section for x between $n- \epsilon$ and n (for very small $\epsilon$ to drop continuously to 0 at n. And there is certainly no reason to require that f not be very complicated between integers.

If you don't think there are more than "hundreds" of such functions (or if you think of functions as limited to "formulas") then you have a very restricted idea of "function".

#### CRGreathouse

Homework Helper
If you don't think there are more than "hundreds" of such functions (or if you think of functions as limited to "formulas") then you have a very restricted idea of "function".
Before there's any confusion, the "hundred" comment is mine (in jest).

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