Isolated points

Let $X \subset \mathbb{R}$. We say a point $x \in X$ is an isolated point if there exists a $\delta > 0$ such that $(t-\delta,t+\delta) \cap X = \emptyset$. It is known that for any such $X$, the set of isolated points of $X$ is at most countable. Note that all sets of isolated points are closd sets, for example $\left\{ \dfrac{1}{n} \colon n=1,2,\ldots \}$ is not closed, since $0$ is a limit points of this set.

Let $\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$ be a time scale of isolated points with $t_k > t_n$ iff $k>n$. Define the bijection $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.

The set $h\mathbb{Z}=\{\ldots,-2h,-h,0,h,2h,\ldots\}$ of multiples of the integers is a time scale.

$\mathbb{T}=\{\ldots,t_{-1},t_0,t_1,\ldots\}$

Generic element $t\in \mathbb{T}$:	For some $n \in \mathbb{Z}, t=t_n$
Jump operator:	$\sigma(t)=\sigma(t_n)=t_{n+1}$
Graininess operator:	$\mu(t)=\mu(t_n)=t_{n+1}-t_n$
$\Delta$-derivative:	$f^{\Delta}(t)=f^{\Delta}(t_n) = \dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$
$\Delta$-integral:
Exponential function: