Can anyone suggest a non-piecewise function within the interval (0,1)?

[rbzima]

I'm just wondering if anyone can think of a function that is contained in (0,1). I've been brainstorming for awhile and I can't think of anything particularly. Furthermore, it would be fantastic if it was not piecewise defined as well. Help would be appreciated!

 

[sutupidmath]

what do u mean it is contained, that for any values of x the values of the function varie from y=0 to y=1 or what?

 

[rbzima]

I'm essentially looking for a function where x is contained in the open interval of (0,1). Someone suggested tan(x) might work, but I would rather not use trigonometric functions.

 

[VietDao29]

You can give any function you think of, and then restrict the domain of the function to be (0, 1), like this:

f: (0, 1) --> R
x |--> x² + x - 5

That means, your f now, is only defined in (0, 1), and undefined everywhere else.

Or, if you want some other nicer function, then square root is a function to look for:

In the reals, the function $$y = \frac{1}{\sqrt{-x(x - 1)}}$$ is only defined on the interval (0, 1), and undefined everywhere else.

 

[rbzima]

Sorry, I should have clarified. I'm trying to prove that there exists a function between (0,1) that is 1-1 and onto, therefore the range of this function must span all y values, and the domain is contained in (0,1).

For example, between (-1,1) f(x) = x/(x^2-1) works, but when I tried converting that into a form that is simply between (0,1), the inverse function looks exceptionally nasty! Using the inverse, I'm basically proving the function is onto, which in turn is proving that the (0,1) ~ R. Proving the function is 1-1 simply requires that the derivative is either always positive or always negative.

 

[HallsofIvy]

One to one and onto? Onto what? If I were talking about a function with domain between 0 and 1 and said it was "onto" the reasonable assumption would be it was "onto" (0, 1). The function y= x does that nicely!

As for a function that is one-to-one and onto from (0, 1) to the set of all real numbers, I would think a variation on tan(x) would work nicely. tan(x) covers all real numbers for x between $-pi/2$ and $\pi/2$ so $f(x)= tan(-\pi/2+ \pi x)$ should work nicely: when x= 0, $-\pi/2+ \pi (0)= -pi/2$. When x= 1, $-pi/2+ \pi (1)= \pi/2$. Unless I have misunderstood what you are looking for, that should be exactly what you want.

 

[rbzima]

One to one and onto? Onto what? If I were talking about a function with domain between 0 and 1 and said it was "onto" the reasonable assumption would be it was "onto" (0, 1). The function y= x does that nicely!

As for a function that is one-to-one and onto from (0, 1) to the set of all real numbers, I would think a variation on tan(x) would work nicely. tan(x) covers all real numbers for x between $-pi/2$ and $\pi/2$ so $f(x)= tan(-\pi/2+ \pi x)$ should work nicely: when x= 0, $-\pi/2+ \pi (0)= -pi/2$. When x= 1, $-pi/2+ \pi (1)= \pi/2$. Unless I have misunderstood what you are looking for, that should be exactly what you want.

You are pretty much the man! Thanks a whole heap. And yes, I was looking for a 1-1 correspondence between (0,1) and R. I might have neglected to mention that.