Spivaks Chain Rule Proof (Image of proof provided) Here is a proof of the chain rule in Spivaks Calculus. Note there is a second page, but I understand it, and this is the meat of the proof. I have a few questions.

$\textbf{1.}$ "It should be intuitively clear that $\phi$ is continuous at $0$." Do we care that it is continuous at zero so we will not have a division by zero since $g(a+h)-g(a)$ is in the denominator and could equal zero? I am not sure I understand why it is continuous at zero. I understand what he was saying but I was always under the impression continuity was when there was no breaks in the graph visually. Here, I am imagining $\phi(h)$ being continuous up to zero, then it jumping to another point when it is zero.

$\textbf{2.}$ At (2),I do not understand what we are trying to do here. We randomly switched to $h$ and are defining continuity I think. The switch back and forth from $k$ to $h$ is confusing me.
The even more intuitive reason why the result should be true is that:

$$\begin{split} (f \circ g)'(a) &amp; = \lim_{h\to0} \frac{f(g(a+h)) - f(g(a)))}{h} \\
  &amp; = \lim_{h\to0} \frac{f(g(a+h)) - f(g(a)))}{g(a+h) - g(a)} \frac{g(a+h) - g(a)}{h} \\
  &amp; = (\lim_{g(a+h)\to g(a)} \frac{f(g(a+h)) - f(g(a)))}{g(a+h) - g(a)}) (\lim_{h\to0}\frac{g(a+h) - g(a)}{h}) \\
  &amp; = f'(g(a)) g'(a)
\end{split}$$

Unfortunately this intuitive proof can fall apart when $g'(a) = 0$ because you may be multiplying and dividing by 0.  So he's creating an auxiliary function which is guaranteed well-defined, and has the limit he wants for the first part of that expression.  Then he's using that function to get around the potential division by 0 problem.

Editorial here.

I have always hated this proof.  I much prefer thinking in terms of the tangent line approximation in which $f(a+h) \approx f(a) + f'(a) h$.  If you define $\approx$ right, this leads to a definition which is fully equivalent to the usual limit definition.  And now the intuitive reason why this is true is that:

$$\begin{split}
f(g(a+h)) &amp; \approx f(g(a)) + f'(g(a)) ( g(a+h) - g(a) ) \\
&amp; \approx f(g(a)) + f'(g(a))(g(a) + g'(a)h - g(a)) \\
&amp; = f(g(a)) + f'(g(a))g'(a)h\end{split}$$

Proving the first approximation does require an admittedly more technical proof.  However it leads to a correct proof of the chain rule that works without change for multi-variable calculus.  As well I think it important for people to really understand the tangent line approximation.
Spivak's is an introductory book, that in theory should be usable by people who have never seen any calculus before, so he is making a long story out of something simple (but profound).   His exposition is idiosyncratic in some places and is not necessarily worth trying to understand in every detail when clearer approaches to the same things are available. I'd suggest ignoring this forced attempt to make $f'(g(x))$ into a ratio, and reading the very short and easily understood proof by linear approximation (see btilly's answer) given in books on analysis such as Rudin.