I know that this is a fairly provocative title so before judging please read what I have to say.
Below is a definition of a derivative taken from Analysis on Manifolds by James Munkres:
Definition: Let $A \subseteq \mathbb{R}$ and let $f : A \to \mathbb{R}$ be any function. Suppose that $A$ contains a neighborhood of the point $a$. We define the derivative of $f$ at $a$ by the equation $$f'(a) = \lim_{t \to 0} \frac{f(a+t) - f(a)}{t}$$ provided that the limit exists.
Now there are a few subtleties to this definition, I will list them below
- The phrase "neighborhood of the point $a$" means an open set in $\mathbb{R}$ that contains $a$ (as opposed to an open set in the subspace $A$ that contains $a$)
- The fact that the definition above states that we need $A$ to contain a neighborhood of $a$ implies that $a$ must be an interior point of $A$. So $a$ must be an interior point of $A$ for $f$ to even meet the criteria needed for $f$ to be differentiable at $a$. (Why not just let $A$ be an open set and be done?)
- We are taking the limit of a function (which is $\frac{f(a+t) - f(a)}{t}$) for which no domain is specified, and a domain really needs to be specified as this quotient wouldn't even be a function without a domain specified.
So taking all of this into account I tried to formulate my own definition of the derivative of a function
My Definition: Let $U \subseteq \mathbb{R}$ be an open set. Let $f : U \to \mathbb{R}$ be any function. Pick $a \in U$ and choose $r > 0$ such that $B(a, r) \subseteq U$. Define $\phi : B(0, r) \setminus \{0\} \to \mathbb{R}$ by $$\phi(t) = \frac{f(a+t) - f(a)}{t}.$$ Then we define the derivative of $f$ at $a$ as $$f'(a) = \lim_{t \to 0} \phi(t)$$ provided that the limit exists.
First of all is my definition correct and equivalent to the conventional definition given by Munkres?
Secondly if my definition is correct why wouldn't the author simply state the above definition? It seems as if a lot of the core detail of the definition has been obfuscated and as a reader it took me quite a while to unpack most of the definition and furthermore without specifying the domain of the quotient function that we're taking the limit of, it doesn't even make sense to take the limit of it.