Two questions on Do Carmo’s definition of variation of a curve

Question

On page 345 of Differential Geometry of Curves and Surfaces by Manfredo Do Carmo there is the following definition of variation of a curve:

Let $\alpha \colon [0, l] \to S$ be a regular parametrised curve, where the parameter $s$ is the arc length. A variation of $\alpha$ is a differentiable map $h \colon [0, l] \times (-\varepsilon, \varepsilon) \subseteq \mathbb{R}^2 \to S$ such that $h(s, 0) = \alpha(s)$ for all $s \in (0, l].$

I have two questions regarding this defintion:

1. Shouldn’t $h(s, 0) = \alpha(s)$ also hold for $s = 0$?
2. What does it mean for $h$ to be differentiable on $[0, l] \times (-\varepsilon, \varepsilon),$ which is not an open set?

Any help will be appreciated.

Answer 1

1. No, because the idea is that this concept is defined for every continuous curve. If we imposed that the equality holds when $s=0$, that would mean that $\alpha$ would be differentiable.
2. So what? Isn't the map $f\colon[-1,1]\longrightarrow\mathbb R$ defined by $f(x)=x$ differentiable?