Let $\gamma : \left\{ \begin{array}{ccc} \mathbb{R} & \to & \mathbb{R} \\ t & \mapsto & (t^2,t^3) \end{array} \right.$ and $\Gamma= \gamma(\mathbb{R})$. Because of the singularity at $(0,0)$, there is no $\mathcal{C}^1$ regular parametrization of $\Gamma$; but there exists such parametrization for $\Gamma_+=\gamma([0,+ \infty[)$. However, $\Gamma_+$ seems not to admit $\mathcal{C}^2$ regular parametrization.
How do you show this result?