Delta mean value theorem

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 $r \leq s$.

References

• Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Theorem $1.67$