# Help finding this function?

• Dec 7th 2009, 04:32 PM
paupsers
Help finding this function?
The idea of this "proof" is very easy to understand.

Let f:[0,1]-->[0,1] be a continuous function. So you have a square one unit across and one unit tall. Draw a function inside of this box so that if you draw a horizontal line through the function anywhere, it will pass through the function 0 times or an even number of times (not infinitely many times).

Remember, the function has to be continuous on [0,1].

I thought for a while that this function doesn't exist, but my professor today confirmed it DOES exist. Has anyone ever heard of a theorem like this, and/or does anyone know a function that satisfies the criteria?
• Dec 7th 2009, 05:47 PM
TKHunny
Why is that hard? Maybe I'm not understanding the requirements. I presume tangent would count as passing through zero times.

Tell me what's wrong with $\displaystyle f(x) = \frac{sin(2k\pi x)}{2}+\frac{1}{2}$ for k an integer and maybe we can understand the task.
• Dec 7th 2009, 06:28 PM
paupsers
Take k=1, for example.

Then
$\displaystyle f(x) = \frac{sin(2\pi x)}{2}+\frac{1}{2}$

So,
$\displaystyle f^{-1}(1)=.25$

In other words, the horizontal line y=1 only intersects the graph of f(x) one time, which is an odd number of times. So that function doesn't work. I need a function so that any horizontal line will cross it finitely many even times or zero times on [0,1], [0,1].

I don't know what you meant about tangent being "zero times."

Do you get what I mean now?
• Dec 7th 2009, 06:38 PM
paupsers
Also consider k=2. Then the horizontal line y=0.5 passes through the function 5 times...
• Dec 8th 2009, 08:20 AM
paupsers
Anyone else have any ideas on this one?
• Dec 8th 2009, 02:16 PM
TKHunny
You will have to make up your mind what you mean. The word "cross" is not well-defined. Do you mean "intersect"? If you do, then I agree with your negative assessment. If you do not mean "intersect", then you have not refuted my functions.
• Dec 8th 2009, 02:43 PM
paupsers
Correct. They cannot intersect.

I don't understand how this is possible, but my professor says it is.
• Dec 10th 2009, 03:03 PM
paupsers
This is the solution, if anyone was interested.

http://www.math.ufl.edu/~pilyugin/Pics/evenpreimage.pdf