# Completely delta differentiable

Let $\Lambda^n$ be an $n$-dimensional time scale. We say that a function $f \colon \Lambda^n \rightarrow \mathbb{R}$ is completely delta differentiable at a point $$t^0=(t_1^0,\ldots,t_n^0) \in \mathbb{T}_1^{\kappa} \times \ldots \times \mathbb{T}_n^{\kappa}$$ if there exist numbers $A_1,\ldots,A_n$ independent of $t=(t_1,\ldots,t_n) \in \Lambda^n$ (but in general dependent on $t^0$)such that for all $t \in U_{\delta}(t^0)$, $$f(t_1^0,\ldots,t_n^0)-f(t_1,\ldots,t_n)= \displaystyle\sum_{i=1}^n A_i(t_i^0-t_i)+\displaystyle\sum_{i=1}^n\alpha_i (t_i^0-t_i),$$ and in addition for each $j \in \{1,\ldots,n\}$ and all $t \in U_{\delta}(t^0)$, $$f(t_1^0,\ldots,t_{j-1}^0,\sigma_j(t_j^0),t_{j+1}^0,\ldots,t_n^0) - f(t_1,\ldots,t_n)=A_j[\sigma_j(t_j^0)-t_j]+\displaystyle\sum_{\stackrel{i=1}{i\neq j}}A_i(t_i^0-t_i)+\beta_{jj}[\sigma_j(t_j^0)-t_j]+\displaystyle\sum_{\stackrel{i=1}{i\neq j}}^n \beta_{ij}(t_i^0-t_i),$$ where $\delta>0$ is sufficiently small, $U_{\delta}(t^0)$ is a $\delta$-neighborhood of $t^0$ in $\Lambda^n$, $\alpha_i=\alpha_i(t^0,t)$ and $\beta_{ij}=\beta_{ij}(t^0,t)$ are defined on $U_{\delta}(t^0)$ such that they are equal to zero at $t=t^0$ and the following two formulas hold: $$\displaystyle\lim_{t \rightarrow t^0}\alpha_i(t^0,t)= 0; i \in \{1,\ldots,n\}$$ and $$\displaystyle\lim_{t \rightarrow t^0} \beta_{ij}(t^0,t)=0; i,j\in\{1,\ldots,n\}.$$