In most if not all elementary calculus books, a function $f : A \rightarrow \mathbb{R}$ with $A \subseteq \mathbb{R}$ is said to be "differentiable'' at a point $a \in A$ if and only if the limit $$ \textrm{lim}_{h \rightarrow 0}\,\,\, \frac{f(a + h) - f(a)}{h} $$ exists, and in which the value of this limit is denoted $f'(a)$.
The problem is, basic statements such as...
"If $f$, $g$ are differentiable at $a$, then $f + g$ is differentiable at $x = a$ and $(f + g)'(a) = f'(a) + g'(a)$."
...or such as...
"If $g$ is differentiable at $a$, and $f$ is differentiable at $g(a)$, then $f \circ g$ is differentiable at $a$ and $(f \circ g)'(a) = f'(g(a))g'(a)$."
...are false under this definition.
One can fix the first statement by adding some assumption about the compatibility of the domains of $f$ and $g$ (e.g., that $a$ not be an isolated point of dom $f$ $\cap$ dom $g$, or that dom $f$, dom $g$ both be open), and likewise one can fix the second statement by adding some ad-hoc small print (that $a$ not be an isolated point of dom $f \circ g$, or that dom $f$ be open, or that $g(a)$ be an interior point of dom $f$, etc). But I'm wondering if the "correct" fix shouldn't be to alter the definition of differentiability to make the above statements automatically correct, with no added small print. Specifically, require as part of the definition of differentiability that $a$ be in the interior of $\textrm{dom}\,\, f$, i.e., that $f$ be defined in a neighbourhood of $a$. (E.g., under this definition no function of domain $[0, \infty)$ would be "differentiable" at $x = 0$.)
Have people seen this definition in use, or seen this point debated? What are people's opinion on this?
And what are the conventions in higher dimensions, i.e., for functions $f : A \subseteq \mathbb{R}^n \rightarrow \mathbb{R}$?
Thanks!
You're right that it is rarely stated clearly, but I think the common assumption when talking about derivatives is that the functions are defined on an open neighborhood of a, as you suggest. I think in most calculus books, they assume adding too many details will confuse students so they leave it out.
I'm taking a light complex analysis class that's aimed at engineers right now, and I'm often frustrated by the professor's unwillingness to use terms like "open set" that are really important parts of definitions of terms we're using. Oftentimes professors do things in the hope of simplifying things that, in my opinion, actually make them more complicated.
Edit: As evidence that this practice is common, in Wikipedia's definition of the derivative they start with this:
"Let f be a real valued function defined in an open neighborhood of a real number a."