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Math Help - continuous curve?

  1. #1
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    continuous curve?

    I'm taking a history of mathematics course and I have a question on my assigment: Give the definition, "the curve g:[0,1] to R^2 (= R x R) is continuous" (the elements of R x R are ordered pairs (x, y) of reals; each such pair represents, via a chosen Cartesian coordinate system, a point in the plane). I know the definition of a continuous function f using the epsilon-delta notation (and much simpler definitions), but how can I define this, a curve with R^2? Is it similar?


    This is also pretty random, but does anyone know if Euclid had a theorem similar to the Intermediate Value Theorem and how he proved it?
    Last edited by MKLyon; November 19th 2007 at 12:34 PM.
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  2. #2
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    In general given a vector-valued function \bold{f}: \mathbb{R}^n \mapsto \mathbb{R}^m we say \bold{f} is continous at (vector not number here) \bold{a}\in \mathbb{R}^n iff |f(\bold{x})-f(\bold{a})|<\epsilon \mbox{ for }|\bold{x}-\bold{a}|<\delta where | ~ | is the 'Euclidean norm' defined for in the following way: "if \bold{a} = (a_1,...,a_n) then |\bold{a}| = \sqrt{a_1^2+...+a_n^2}". Does that make sense?


    I wish I was taking History of Math couse.
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  3. #3
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    We can think of your problem like I stated above or in a different way.

    We have \bold{f}:\mathbb{R}\mapsto \mathbb{R}^2 so we can view \bold{f} = (f_1,f_2), i.e. the components of the vector-valued function. And each f_1,f_2 is continous. Maybe this is an easier way to think of this.
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