# Parametric Curve of y=|x| and differentiability

#### Maidenas

##### New member
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.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hi Maidenas, welcome to MHB!

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

#### Maidenas

##### New member
Indeed !! thank you very much my friend )

#### Maidenas

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

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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.