Consider the curve $C=\bigl\{(t,|t|): t \in \mathbb R\bigr\}$.Show that if $\alpha:\bigr]a,b\bigr[ \to \mathbb R$ is a differentiable curve whose trace lies in C and $t_0 \in \bigr]a,b\bigr[$ is such that $\alpha(t_0)=(0,0)$ then $\alpha'(t_0)=(0,0)$.
How can one assure that there will be points in the neighborhood of $t_0$ such that $\alpha_1$ is negative and $\alpha_1$ is non-negative? If one can assure that, we can verify that $\alpha'(t_0)=(0,0)$ by studying the derivative of $\alpha$ at $t_0$ by the definition of limit and consider both cases with $\alpha_1$ non-negative or $\alpha_1$ positive.
Let $\alpha=(\alpha_1,|\alpha_1|)$ and $\alpha'(t_0)=(a,b).$
As $t\to t_0,$
$$\alpha_1(t)=a(t-t_0)+o(t-t_0)\quad\text{and}\quad|\alpha_1(t)|=b(t-t_0)+o(t-t_0)$$
hence $$b\left(t-t_0\right)=|a|\left|t-t_0\right|+o(t-t_0),$$ or equivalently $$b=\lim_{t\to t_0}\left(|a|\frac{\left|t-t_0\right|}{t-t_0}\right),$$ which implies $$a=b=0.$$