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 clasical derivative and the integrals reduce to the Riemann integral.

$\mathbb{T}=\mathbb{R}$
Generic element $t \in \mathbb{T}$: $t=t$
Forward jump: $\sigma(t)=t$
Graininess: $\mu(t)=0$
Backward jump: $\rho(t)=t$
Backward graininess: $\nu(t)=0$
$\Delta$-derivative $$f^{\Delta}(t)=\lim_{h\rightarrow 0} \dfrac{f(t+h)-f(t)}{h}=f'(t)$$
$\nabla$-derivative $$f^{\nabla}(t) =\lim_{h \rightarrow 0} \dfrac{f(t)-f(t-h)}{h}= f'(t)$$
$\Delta$-integral $$\int_s^t f(\tau) \Delta \tau = \int_s^t f(\tau) d\tau$$
$\nabla$-derivative $$\int_s^t f(\tau) \nabla \tau = \int_s^t f(\tau) d\tau$$
$\Delta$-exponential $\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\ &\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\ &= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) \end{array}$
$\nabla$-exponential $\hat{e}_p(t,s)=\exp \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
$\mathrm{sin}_p(t,s)$
$\mathrm{\sin}_1(t,0)$ $\sin(t)$
$\mathrm{\cos}_p(t,s)$ $\cos \left( \displaystyle\int_s^t p(\tau) d\tau \right)$
$\mathrm{\cos}_1(t,0)$ $\cos(t)$
Hilger circle
Laplace transform $\mathscr{L}_{\mathbb{R}}\{f\}(z;s)=\displaystyle\int_0^{\infty} f(\tau) e^{-z\tau} d\tau$
Gamma function