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 $C_{\mathrm{rd}}^n(\mathbb{T},X)$.
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=Properties=
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[[Continuous implies rd-continuous]]<br />
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[[rd-continuous implies regulated]]<br />
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[[Forward jump is rd-continuous]]<br />
  
 
=References=
 
=References=
* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Regulated function|next=findme}}: Definition $1.58$
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* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Regulated|next=Continuous implies rd-continuous}}: Definition $1.58$
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* {{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