Time-scale calculus

Definition

A time scale is a nonempty closed subset $\mathbb{T}$ of $\R$. The forward jump operator $\sigma \colon \mathbb{T} \to \mathbb{T}$ is $\sigma(t) = \inf \set{s \in \mathbb{T} : s > t}$, with $\sigma(t) = t$ when $t$ is the largest element of $\mathbb{T}$. The graininess is $\mu(t) = \sigma(t) - t$. Let $\mathbb{T}^\kappa$ be $\mathbb{T}$ without its largest element $m$ if $m$ exists and $\sup \set{s \in \mathbb{T} : s < m} < m$; otherwise $\mathbb{T}^\kappa = \mathbb{T}$. Let $f \colon \mathbb{T} \to \R$ and $t \in \mathbb{T}^\kappa$. The function $f$ is delta differentiable at $t$, with delta derivative $f^\Delta(t) \in \R$, if for every $\varepsilon > 0$ there is $\delta > 0$ such that $$\abs{f(\sigma(t)) - f(s) - f^\Delta(t) \, (\sigma(t) - s)} \le \varepsilon \abs{\sigma(t) - s} \quad \text{for all } s \in \mathbb{T} \text{ with } \abs{t - s} < \delta.$$ Time-scale calculus is the study of delta derivatives, of the matching integrals, and of dynamic equations built from them.