Difference between revisions of "Delta mean value theorem"
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(Created page with "==Theorem== Let $\mathbb{T}$ be a time scale and let $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ be pre-differentiable with $D$. Then $|f^{\Delta}(t)| \leq g^{\Delt...") |
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==Theorem== | ==Theorem== | ||
| − | Let $\mathbb{T}$ be a [[time scale]] and let $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ be [[pre-differentiable]] with $D$. | + | Let $\mathbb{T}$ be a [[time scale]] and let $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ be [[pre-differentiable]] with $D$. If for all $t \in \mathbb{T}$, $|f^{\Delta}(t)| \leq g^{\Delta}(t)$, where $f^{\Delta}$ denotes [[delta derivative]], then |
$$|f(s)-f(r)| \leq g(s)-g(r)$$ | $$|f(s)-f(r)| \leq g(s)-g(r)$$ | ||
for all $r,s \in \mathbb{T}$ with $r \leq s$. | for all $r,s \in \mathbb{T}$ with $r \leq s$. | ||
Latest revision as of 00:07, 5 January 2017
Theorem
Let $\mathbb{T}$ be a time scale and let $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ be pre-differentiable with $D$. If for all $t \in \mathbb{T}$, $|f^{\Delta}(t)| \leq g^{\Delta}(t)$, where $f^{\Delta}$ denotes delta derivative, then $$|f(s)-f(r)| \leq g(s)-g(r)$$ for all $r,s \in \mathbb{T}$ with $r \leq s$.
Proof
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Theorem $1.67$
