Real numbers

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The set $\mathbb{R}$ of real numbers is a time scale. In this time scale, all derivatives reduce to the classical derivative and the integrals reduce to the classical integral.

$\mathbb{T}=\mathbb{R}$
Forward jump: $\sigma(t)=t$ derivation
Forward graininess: $\mu(t)=0$ derivation
Backward jump: $\rho(t)=t$ derivation
Backward graininess: $\nu(t)=0$ derivation
$\Delta$-derivative $f^{\Delta}(t)=\displaystyle\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$ derivation
$\nabla$-derivative $f^{\nabla}(t) =\displaystyle\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$ derivation
$\Delta$-integral $\displaystyle\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$ derivation
$\nabla$-derivative $\displaystyle\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$ derivation
$h_k(t,s)$ $h_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$\hat{h}_k(t,s)$ $\hat{h}_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$g_k(t,s)$ $g_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$\hat{g}_k(t,s)$ $\hat{g}_k(t,s)=\dfrac{(t-s)^k}{k!}$ derivation
$e_p(t,s)$ $e_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
Gaussian bell $\mathbf{E}(t)=e^{-\frac{t^2}{2}}$ derivation
$\mathrm{sin}_p(t,s)=$ $\sin_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\mathrm{\sin}_1(t,s)$ $\sin_1(t,s)=\sin(t-s)$ derivation
$\widehat{\sin}_p(t,s)$ $\widehat{\sin}_p(t,s)=\sin\left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\mathrm{\cos}_p(t,s)$ $\cos_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ derivation
$\mathrm{\cos}_1(t,s)$ $\cos_1(t,s)=\cos(t-s)$ derivation
$\widehat{\cos}_p(t,s)$ $\widehat{\cos}_p(t,s)=\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$ 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
$\widehat{\cosh}_p(t,s)$ $\widehat{\cos}_p(t,s)=$ derivation
Gamma function $\Gamma_{\mathbb{R}}(x,s)=\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{\mathbb{T}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\mathbb{T}}(t)=$ derivation
Laplace transform $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$ 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