Isolated points

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Let $\mathbb{T}$ be a time scale. We say that $\mathbb{T}$ is a time scale of isolated points if there exists $\epsilon > 0$ such that for all $t \in \mathbb{T}$, $\mu(t) \geq \epsilon$. 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 $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.

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


Examples of time scales of isolated points