Difference between revisions of "Rd-continuous"

From timescalewiki
Jump to: navigation, search
 
(4 intermediate revisions by the same user not shown)
Line 3: Line 3:
 
=Properties=
 
=Properties=
 
[[Continuous implies rd-continuous]]<br />
 
[[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 function|next=findme}}: Definition $1.58$
+
* {{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