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Continuous function from (0,1) onto [0,1]?

  1. Nov 9, 2012 #1
    I know that there does not exist a continuos function from [0,1] onto (0,1) because the image of a compact set for a continous function f must be compact, but isn't it also the case that the inverse image of a compact set must be compact? and a set in R is compact iff its closed and bounded right?
     
  2. jcsd
  3. Nov 9, 2012 #2

    jgens

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    Gold Member

    First consider the continuous surjection [itex]f:(0,1) \rightarrow [0,1][/itex] defined as follows:
    [tex]
    f(x) = \left\{
    \begin{array}{lcl}
    0 & : & x \in \left(0,\frac{1}{4}\right)\\[0.3em]
    2x-\frac{1}{2} & : & x \in \left[\frac{1}{4},\frac{3}{4}\right]\\[0.3em]
    1 & : & x \in \left(\frac{3}{4},1\right)
    \end{array}
    \right.
    [/tex]
    Now to answer your other questions ...

    Not all continuous maps satisfy this property. Those that do are called 'proper'.

    Assuming that you give [itex]\mathbb{R}[/itex] its usual topology and assuming you mean bounded with respect to the Euclidean metric, then yes.
     
    Last edited: Nov 9, 2012
  4. Nov 10, 2012 #3

    lavinia

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    Science Advisor

    map any set with any topology to a point. This map is continuous and its image is compact.
     
  5. Nov 12, 2012 #4
    (1/2) (sin(2pi x)+1) maps (0,1) onto [0,1].

    Clearly the problem is that this function is not injective. Same problem with the example by jgens.

    Does there exist an injective continuous function mapping (0,1) onto [0,1]? Assume there is, and suppose f(a)=0 and f(b)=1. WLOG assume b>a and let e>0 be small enough so that b+e<1. Since 1 is the max value of f, f(b+e) is strictly between 0 and 1. By the IVT, there exists c between a and b such that f(c)=f(b+e). So f can't be injective after all.

    More generally if f is injective and continuous from an interval of R into R, then it must be monotonic and its inverse must be continuous as well.
     
  6. Nov 12, 2012 #5

    Bacle2

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    Science Advisor

    You can also use a result that a continuous bijection f: X--->Y , between X compact

    and Y Hausdorff , is a homeomorphism. And (0,1) and [0,1] are not homeomorphic

    for many reasons: [0,1] is compact and (0,1) is not, or (0,1) is 1-connected and

    [0,1] is not -- e.g., delete the endpoints of [0,1], and the space remains connected

    ( I think k-connectedness is also called the Euler number). This also shows there are

    no continuous bijections between (0,1) and [0,1) (because [0,1) is not 1-connected;

    remove 0, and it remains connected.)
     
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