Difference between revisions of "Delta derivative at right-scattered"
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$$f^{\Delta}(t)=\dfrac{f(\sigma(t))-f(t)}{\mu(t)},$$ | $$f^{\Delta}(t)=\dfrac{f(\sigma(t))-f(t)}{\mu(t)},$$ | ||
where $f^{\Delta}$ denotes the [[delta derivative]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]]. | where $f^{\Delta}$ denotes the [[delta derivative]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]]. | ||
| + | |||
| + | ==Proof== | ||
| + | |||
| + | ==References== | ||
| + | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta differentiable implies continuous|next=Delta derivative at right-dense}}: Theorem 1.16 (ii) | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 15:18, 21 January 2023
Theorem
Let $\mathbb{T}$ be a time scale. Let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be continuous and right-scattered at $t \in \mathbb{T}$. Then $$f^{\Delta}(t)=\dfrac{f(\sigma(t))-f(t)}{\mu(t)},$$ where $f^{\Delta}$ denotes the delta derivative, $\sigma$ denotes the forward jump, and $\mu$ denotes the forward graininess.
Proof
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Theorem 1.16 (ii)
