Difference between revisions of "Rd-continuous"
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Let $\mathbb{T}$ be a [[time scale]] and $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[regulated function]]. We say that $f$ is rd-continuous if for any [[dense point|right dense]] point $t \in \mathbb{T}$, $f(t) = \displaystyle\lim_{\xi \rightarrow t^+} f(\xi)$. In other words, $f$ is rd-continuous if it is regulated and continuous at right dense points. The notation $C_{\mathrm{rd}}(\mathbb{T},X)$ denotes the set of rd-continuous functions $g \colon \mathbb{T} \rightarrow X$. We denote the set of rd-continuous functions that are $n$-times [[delta derivative|delta differentiable]] by the notation $C_{\mathrm{rd}}^n(\mathbb{T},X)$. | Let $\mathbb{T}$ be a [[time scale]] and $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a [[regulated function]]. We say that $f$ is rd-continuous if for any [[dense point|right dense]] point $t \in \mathbb{T}$, $f(t) = \displaystyle\lim_{\xi \rightarrow t^+} f(\xi)$. In other words, $f$ is rd-continuous if it is regulated and continuous at right dense points. The notation $C_{\mathrm{rd}}(\mathbb{T},X)$ denotes the set of rd-continuous functions $g \colon \mathbb{T} \rightarrow X$. We denote the set of rd-continuous functions that are $n$-times [[delta derivative|delta differentiable]] by the notation $C_{\mathrm{rd}}^n(\mathbb{T},X)$. | ||
| + | |||
| + | =Properties= | ||
| + | [[Continuous implies rd-continuous]]<br /> | ||
| + | [[rd-continuous implies regulated]]<br /> | ||
| + | [[Forward jump is rd-continuous]]<br /> | ||
=References= | =References= | ||
| − | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Regulated | + | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Regulated|next=Continuous implies rd-continuous}}: Definition $1.58$ |
| + | * {{PaperReference|Functional series on time scales|2008|Dorota Mozyrska|author2=Ewa Pawluszewicz|prev=Regulated|next=Pre-differentiable}} | ||
Latest revision as of 14:53, 21 October 2017
Let $\mathbb{T}$ be a time scale and $f \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regulated function. We say that $f$ is rd-continuous if for any right dense point $t \in \mathbb{T}$, $f(t) = \displaystyle\lim_{\xi \rightarrow t^+} f(\xi)$. In other words, $f$ is rd-continuous if it is regulated and continuous at right dense points. The notation $C_{\mathrm{rd}}(\mathbb{T},X)$ denotes the set of rd-continuous functions $g \colon \mathbb{T} \rightarrow X$. We denote the set of rd-continuous functions that are $n$-times delta differentiable by the notation $C_{\mathrm{rd}}^n(\mathbb{T},X)$.
Properties
Continuous implies rd-continuous
rd-continuous implies regulated
Forward jump is rd-continuous
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Definition $1.58$
- Dorota Mozyrska and Ewa Pawluszewicz: Functional series on time scales (2008)... (previous)... (next)
