The following definition of the derivative of a function is taken from Analysis on Manifolds by James Munkres:
Now what I'm confused about is what definition of a limit is being used above. There are two definitions that I'm familiar with from analysis
Definition (Limit of a function): Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Suppose that $E \subseteq X$. Let $f : E \to Y$ and suppose that $p$ is a limit point of $E$. We say that $$\lim_{x \to p} f(x) = q$$ is there exists a $q \in Y$ such that for all $\epsilon > 0$ there exists a $\delta > 0$ such that $$0 < d_X(x, p) < \delta \implies d_Y((f(x), q) < \epsilon$$ for any $x \in B_{(X, d_x)}(p, \delta)$
Definition (Limit of a sequence): We say that the sequence $\{x_n\}_{n \in \mathbb{N}} \subseteq (X, d)$ converges to $x' \in X$ and write $$\lim_{n \to \infty} x_n = x'$$ if for any $\epsilon > 0$ there exists a natural number $N > 0$ such that $d(x_n, x') < \epsilon$ for all $n > N$
I'm sure that the author must be using the limit (of a function) in the definition of a derivative. Now what I'm confused about is that there is no domain specified for the new function $f : \ ? \to \mathbb{R}$ defined by $$f(t) = \frac{\phi(a+t) -\phi(a)}{t}$$ and also I have no way to tell if $0$ is actually a limit point for the domain of this new function.
What is the domain of this new function (which is being used in the definition of the derivative)?
Update: Based on the comments below I understand the following: If $A \subseteq \mathbb{R}$ and $\phi : A \to \mathbb{R}$ is a function and $A$ contains a neighborhood $U$ of the point $a$ then we define the derivative of $\phi$ at $a$ by the equation $$\phi'(a) = \lim_{t \to 0} f(t)$$ where $f : B(a, r) \to \mathbb{R}$ is defined by $$f(t) = \frac{\phi(a+t) -\phi(a)}{t}$$ and $r$ is some real number greater than $0$ such that the open ball $B(a, r) \subseteq U$.
Now the only problem with the above definition of a derivative is that by the definition of a limit of a function we need $0$ to be a limit point of $B(a, r)$ but it may be the case that $0$ is not a limit point of $B(a, r)$. (To see why note that by the definition of $B(a, r)$ we have $B(a, r) = (a-r, a+r)$ and we could have $a = 50$ and $r=1$ for example)
So how exactly does one apply the definition of a limit of a function to define the definition of the derivative of a function?