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Math Help - real analysis question

  1. #1
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    real analysis question

    this is a problem from kenneth ross's book "elementary analysis: the theory of calculus" in section 21 (metric spaces):

    we say a function f maps a set E onto a set F provided f(E)=F.
    1. show that there is a continuous function mapping the unit square {(x_1,x_2) in R^2: 0<=x_1<=1, 0<=x_2<=1} onto [0,1].

    2. do you think there is a continuous function mapping [0,1] onto the unit square?
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  2. #2
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    Quote Originally Posted by squarerootof2 View Post
    this is a problem from kenneth ross's book "elementary analysis: the theory of calculus" in section 21 (metric spaces):

    we say a function f maps a set E onto a set F provided f(E)=F.
    1. show that there is a continuous function mapping the unit square {(x_1,x_2) in R^2: 0<=x_1<=1, 0<=x_2<=1} onto [0,1].
    Obvious: for example, f(x_1,x_2) = x_1.

    Quote Originally Posted by squarerootof2 View Post
    2. do you think there is a continuous function mapping [0,1] onto the unit square?
    Not so obvious! You are looking for a space-filling curve.
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  3. #3
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    i'm just curious, but is there a way to explicitly define a space-filling curve as a function? i've been looking around and it seems like everyone just uses the computer to draw the curves...
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    Quote Originally Posted by squarerootof2 View Post
    i'm just curious, but is there a way to explicitly define a space-filling curve as a function? i've been looking around and it seems like everyone just uses the computer to draw the curves...
    Here are two pages that may help with that question.
    Hilbert Curve -- from Wolfram MathWorld

    Space-Filling Function -- from Wolfram MathWorld
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