Delta derivative on a time scale

Definition

Let $\mathbb{T}$ be a time scale, that is, a nonempty closed subset of $\R$. Let $\sigma(t) = \inf \set{s \in \mathbb{T} : s > t}$ for $t \in \mathbb{T}$, with $\sigma(t) = t$ when $t = \max \mathbb{T}$. Let $\mathbb{T}^\kappa$ be $\mathbb{T}$ without $m = \max \mathbb{T}$ if this maximum exists and $\sup \set{s \in \mathbb{T} : s < m} < m$, and $\mathbb{T}^\kappa = \mathbb{T}$ otherwise. Let $f \colon \mathbb{T} \to \R$ and $t \in \mathbb{T}^\kappa$. A number $f^\Delta(t) \in \R$ is the delta derivative of $f$ at $t$ if the following holds. For every $\varepsilon > 0$ there is $\delta > 0$ such that $$\abs{f(\sigma(t)) - f(s) - f^\Delta(t)\,(\sigma(t) - s)} \le \varepsilon \abs{\sigma(t) - s}$$ for all $s \in \mathbb{T}$ with $\abs{t - s} < \delta$. Then $f$ is called delta differentiable at $t$.