The most general way to define the derivative of a real function is given as follows:
Definition$\;\mathbf{1}$. Let $X \subset \mathbb{R}$, $f: X \rightarrow \mathbb{R}$ and $a \in X$ a cluster point (point of accumulation or limit point) of $X$. We say that $f$ is differentiable at $a$ when exists the limit $$ f'(a):=\lim_{x \to a} \frac{f(x)-f(a)}{x-a}. \tag{1} $$ In affirmative case, the limit $f'(a)$ is said the derivative of $f$ at $a$.
Remark$\;\mathbf{1}.$ In Defintion $1$ the point $a \in X$ must be a cluster point of $X$ in order to ensure that the limit in $(1)$, if exists, is well-defined and unique.
On the other hand, for instance, in Definition $6.1.1$ (page $162$) of $[1]$, Definition $1$ is considered for $X=I$ an interval. The reason for that, according the authors, is: the concept is most naturally apparent for functions defined on intervals. However, when $X$ is an interval is not considered $a \in X$ as a cluster point.
Question. Regarding this last affirmation: is this possible since every point in any interval is a cluster point?
$[1]$ Bartle, R. and Sherbert, S. Introduction to real analysis. 4th ed.
Given $X \subset \mathbb{R}$, denote by $X':=\{ x \in \mathbb{R} \: ; \: x \: \text{is a cluster point of} \: X\}$.
For $a,b \in \mathbb{R}$ with $a<b$, let $I \subset \mathbb{R}$ be any of the following bounded intervals: $(a,b), (a,b], [a,b)$ and $[a,b]$. According item $i)$ of Example $2.1.1$ of $[2]$, we have $I'=[a,b]$. Thus, in this case, since $I \subset [a,b]=I'$, we conclude that every point of $I$ is a cluster point of $I$.
Now, if $I \subset \mathbb{R}$ is one of the unbounded intervals $(a, \infty)$ or $[a, \infty)$, then according item $ii)$ of Example $2.1.1$ of $[2]$, we have $I'=[a, \infty)$. Thus, $I \subset [a, \infty)=I'$ and every point of $I$ is a cluster point of $I$.
Similarly, if $I \subset \mathbb{R}$ is one of the unbounded intervals $(-\infty,b)$ or $(-\infty,b]$, by item $iii)$ of Example $2.1.1$ of $[2]$, every point of $I$ is a cluster point of $I$.
In any case, namely for bounded or unbounded intervals, we can conclude that every point of any interval $I \subset \mathbb{R}$ is a cluster point of $I$.
$[2]$ NAIR, M. Thamban; NAIR. Calculus of one variable. Second Edition. Springer International Publishing, $2021$.