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?

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 for k an integer and maybe we can understand the task.

Take k=1, for example.

Then


So,

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?

Also consider k=2. Then the horizontal line y=0.5 passes through the function 5 times...

Anyone else have any ideas on this one?

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.

Correct. They cannot intersect.

I don't understand how this is possible, but my professor says it is.

This is the solution, if anyone was interested.

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