# 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 $\pi \colon \mathbb{T} \rightarrow \mathbb{Z}$, $\pi(t_k)=k$.

 Generic element $t \in \mathbb{T}$: for some $n \in \mathbb{Z}$, $t=t_n$ Forward jump: $\sigma(t_n)=t_{n+1}$ derivation Forward graininess: $\mu(t_n)=t_{n+1}-t_n$ derivation Backward jump: $\rho(t_n)=t_{n-1}$ derivation Backward graininess: $\nu(t_n)=t_{n}-t_{n-1}$ derivation $\Delta$-derivative $f^{\Delta}(t_n)=\dfrac{f(t_{n+1})-f(t_n)}{t_{n+1}-t_n}$ derivation $\nabla$-derivative $f^{\nabla}(t_n)=\dfrac{f(t_n)-f(t_{n-1})}{t_{n}-t_{n-1}}$ derivation $\Delta$-integral $\displaystyle\int_{s}^{t} f(\tau) \Delta \tau= \left\{ \begin{array}{ll} -\displaystyle\sum_{k=\pi(t)}^{\pi(s)-1} \mu(t_k)f(t_k) &; ts \end{array}\right.$ derivation $\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=\left\{ \begin{array}{ll} -\displaystyle\sum_{k=\pi(t)+1}^{\pi(s)} \nu(t_k)f(t_k) &; ts \end{array} \right.$ derivation $h_k(t,s)$ $h_k(t,s)=$ derivation $\hat{h}_k(t,s)$ $\hat{h}_k(t,s)=$ derivation $g_k(t,s)$ $g_k(t,s)=$ derivation $\hat{g}_k(t,s)$ $\hat{g}_k(t,s)=$ derivation $e_p(t,s)$ $e_p(t,s)=$ derivation $\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=$ derivation Gaussian bell $\mathbf{E}(t)=$ derivation $\mathrm{sin}_p(t,s)=$ $\sin_p(t,s)=$ derivation $\mathrm{\sin}_1(t,s)$ $\sin_1(t,s)=$ derivation $\widehat{\sin}_p(t,s)$ $\widehat{\sin}_p(t,s)=$ derivation $\mathrm{\cos}_p(t,s)$ $\cos_p(t,s)=$ derivation $\mathrm{\cos}_1(t,s)$ $\cos_1(t,s)=$ derivation $\widehat{\cos}_p(t,s)$ $\widehat{\cos}_p(t,s)=$ derivation $\sinh_p(t,s)$ $\sinh_p(t,s)=$ derivation $\widehat{\sinh}_p(t,s)$ $\widehat{\sinh}_p(t,s)=$ derivation $\cosh_p(t,s)$ $\cosh_p(t,s)=$ derivation $\widehat{\cosh}_p(t,s)$ $\widehat{\cosh}_p(t,s)=$ derivation Gamma function $\Gamma_{\mathbb{T}_{\mathrm{iso}}}(x,s)=$ derivation Euler-Cauchy logarithm $L(t,s)=$ derivation Bohner logarithm $L_p(t,s)=$ derivation Jackson logarithm $\log_{\mathbb{T}_{\mathrm{iso}}} g(t)=$ derivation Mozyrska-Torres logarithm $L_{\mathbb{T}_{\mathrm{iso}}}(t)=$ derivation Laplace transform $\mathscr{L}_{\mathbb{T}_{\mathrm{iso}}}\{f\}(z;s)=$ derivation Hilger circle derivation

## Examples of time scales

$\Huge\mathbb{R}$
Real numbers
$\Huge\mathbb{Z}$
Integers
$\Huge{h\mathbb{Z}}$
Multiples of integers
$\Huge\mathbb{Z}^2$
Square integers
$\Huge\mathbb{H}$
Harmonic numbers
$\Huge\mathbb{T}_{\mathrm{iso}}$
Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$
nth root numbers
$\Huge\mathbb{P}_{a,b}$
Evenly spaced intervals
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q>1$
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
$\Huge\mathcal{C}$
Cantor set