calculus - Rudin's Chain Rule

Rudin's chain rule theorem goes like this:

Suppose $f$ is continuous on ${[a,b]}$, $f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which contains the range of $f$, and $g$ is differentiable at the point $f(x)$. If $$h(t)=g(f(t)) \quad (a\le t \le b)$$ then $h$ is differentiable at $x$, and $$h'(x)=g'(f(x))f'(x)$$

First note that the interval $I$ is closed by Rudin's definition (he calls an open interval $(a,b)$ a segment). Also, Rudin defines differentiability to be in closed intervals, taking the left/right derivative at endpoints.

Since the interval $I$ taken affects the derivative $g'$ (for example, let $g(x) = |x|$, then the derivative can be $1$ or $-1$), I'm wondering if it's true that taking any interval $I$ containing the range of $f$ in which $g$ is (perhaps only left/right) differentiable at $f(x)$ is ok. And in case $g$ is only left-differentiable at $f(x)$ in $I$, we just take the left derivative to be $g'$. Will $h'(x)$ always be the same regardless of which interval $I$ we pick?

Lastly, aside to the question, I wonder if it is a good idea to define differentiability on closed intervals like Rudin does. It does seem to make theorems like this one more complicated. I've found many online sources like Wikipedia defining it on open sets only, but Rudin's book is considered a classic and Terence Tao also defines it the same way. There seems to be no consensus on this issue, so is there a "best practice"?