# Thread: Parametric Curve of y=|x| and differentiability

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.

2.

3. 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.

Indeed !! thank you very much my friend )

if we want the curve to be infinietly differentiable what function we could choose??

6. Originally Posted by Maidenas
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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