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  1. MHB Apprentice

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    #1
    Hello to everyone, this is my first post and a have a question about an exercise.
    It says

    Show that there is (exists) a parametric curve $$\gamma :\Bbb{R}\rightarrow \Bbb{R}^2 $$ which is differentiable such that $$\gamma ( \Bbb{R})=\left\{(x,|x|),x \in \Bbb{R}\right\}$$

    Can be this true? That exersice i noticed from other years notes that he is always putting it without solving it.

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    Klaas van Aarsen's Avatar
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    #2
    Hi Maidenas, welcome to MHB!

    How about the parametrization
    $$\gamma: t\mapsto (t^3,|t^3|)$$
    It means the curve is not regular.
    That is, its derivative is (0,0) at 0.

  4. MHB Apprentice

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    #3 Thread Author
    Indeed !! thank you very much my friend )

  5. MHB Apprentice

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    #4 Thread Author
    if we want the curve to be infinietly differentiable what function we could choose??

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    #5
    Quote Originally Posted by Maidenas View Post
    if we want the curve to be infinitely differentiable what function we could choose??
    I don't think that is possible.
    An infinitely differentiable curve is smooth, meaning it cannot have an angle in it.

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