Difference between revisions of "Continuous"

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 rd-continuous if it is continuous at right-dense points of $\mathbb{T}$ and its left-sided limits exist at left-dense points of $\mathbb{T}$. We use the notation $C_{\mathrm{rd}}(\mathbb{T},X)$ to denote the set of rd-continuous functions $f \colon \mathbb{T} \rightarrow X$. Let $n$ be a positive integer, then notation $C_{\mathrm{rd}}^n$ denotes the set of rd-continuous functions which are $n$-times $\Delta$-differentiable.