I have just start studying about differential geometry and, after reading some basic theory about curves, the following question has arisen :
Let $\alpha:[a,b]\to\mathbb{R}^2$ an injective differentiable map such that $M\geq|\alpha'(s)|\neq0 \;\forall s \in [a,b]$ for some $M>0$. Is it true that for every $p\in C=\alpha([a,b])$ there exists $U_p$ an open neighbourhood of $p$ in $C$ (with the usual restricted topology of $\mathbb{R}^2$) and $\varphi_p:]a_p,b_p[\to\mathbb{R}$ such that $U_p=\{(s,\varphi_p(s):s\in]a_p,b_p[\}$?
Apparently, as we can see in Is a smooth curve locally a graph? the answer is postive but, could we cover the curve with finite many $\varphi_p$?
For any $p$ on the curve you can consider $U_p \subset C$. $U_p$ is an open neighbourhood of $p$ in the restriction topology. Obviously, $\bigcup_{p\in C} U_p = C$, so $\{U_p\}_{p \in C}$ is an open cover of $C$. $C$ is compact as a continuous image of $[a,b]$. Hence there exists a finite subcover $\{U_{p_1}, \dots, U_{p_n}\}$.
If you still have any questions, I will be happy to answer them.