Delta derivative
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Revision as of 02:37, 18 May 2014 by Tom (talk | contribs) (Created page with "Let $\mathbb{T}$ be a time scale. Define $\mathbb{T}^{\kappa} := \mathbb{T} \setminus \sup \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. We define the del...")
Let $\mathbb{T}$ be a time scale. Define $\mathbb{T}^{\kappa} := \mathbb{T} \setminus \sup \mathbb{T}$. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$. We define the delta-derivative of $f$ to be the function $f^{\Delta} \colon \mathbb{T}^{\kappa} \rightarrow \mathbb{R}$ by the formula $$f^{\Delta}(t) := \left\{ \begin{array}{ll} \dfrac{f(\sigma(t))-f(t)}{\mu(t)} &\colon \mu(t) > 0 \\ \displaystyle\lim_{s \rightarrow t} \dfrac{f(s) - f(t)}{s-t} &\colon \mu(t) = 0. \end{array} \right.$$
