Difference between revisions of "Continuous"
m (Tom moved page Continuity to Continuous) 

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Latest revision as of 23:28, 4 January 2017
Let $X \subset \mathbb{R}$ and let $t \in X$. We say that a function $f \colon X \rightarrow \mathbb{R}$ is continuous at $t$ if for every $\epsilon >0$ there exists $\delta >0$ so that for all $s \in (t\delta,t+\delta) \bigcap X$, $f(t)f(s)<\epsilon$.
Let $\mathbb{T}$ be a time scale. We say that a function $f \colon \mathbb{T} \rightarrow \mathbb{R}$ is rdcontinuous if it is continuous at rightdense points of $\mathbb{T}$ and its leftsided limits exist at leftdense points of $\mathbb{T}$. We use the notation $C_{\mathrm{rd}}(\mathbb{T},X)$ to denote the set of rdcontinuous functions $f \colon \mathbb{T} \rightarrow X$. Let $n$ be a positive integer, then notation $C_{\mathrm{rd}}^n$ denotes the set of rdcontinuous functions which are $n$times $\Delta$differentiable.