In the third last line, it is said that because each $g_i$ is continuously differentiable at $a$ then the constructed function $g$ is also differentiable at $a$.
I do not see why "continuously" differentiable is need cuz I think a function is differentiable if and only if each of its components is differentiable (theorem 2-3(3)).
It seems that I am missing something very obvious
Edit: theorem 2-8 is
Yes, you're right, you can drop the "continuously differentiable" hypothesis. Theorem $2$-$9$ is an application of the chain rule (Theorem $2$-$2$) and Theorem $2$-$2(3)$ in a special case. So, you can write the theorem as:
Let $g^1, \dots g^m: \Bbb{R}^n \to \Bbb{R}$ be functions which are differentiable at $a$, and define $g : \Bbb{R}^n \to \Bbb{R}^m$ by $g= (g^1, \dots g^m)$. Suppose $f: \Bbb{R}^m \to \Bbb{R}$ is differentiable at $g(a)$. Then $f \circ g$ is differentiable at $a$ (which means all the partial derivatives exist by Theorem $2$-$7$), and \begin{align} D_i(f \circ g)(a) &= \sum_{j=1}^m (D_jf)(g(a)) \cdot (D_ig^j)(a) \end{align}
However, if you read the previous paragraph, Spivak says
"With Theorem $2$-$8$ to provide differentiable functions, and Theorem $2$-$7$ to provide their derivatives, the chain rule may therefore seem almost superfluous. However, it has an extremely important corollary concerning partial derivatives."
So, yes, the hypotheses of Theorem $2$-$9$ are not the weakest ones you can impose, but Spivak explicitly mentions that it is "an extremely important corollary". He says this because many functions you may encounter typically in the beginning, for example the kind in Problem $2$-$28$ or Problems $2$-$17$ to $2$-$20$ mostly satisfy the special hypothesis of continuous differentiability, and some are even infinitely continuously differentiable (which in those problems, is particularly easy to check) which means calculating partial derivatives becomes reduced to a mechanical procedure (rather than having to directly use the limit definition).
