Multivariable Taylor expansion theory is:
Given open set $D \in \mathbb{R}^n$ $ (n \geq 2)$ , $f : D \rightarrow \mathbb R^p, (p \geq 1)$ and $f \in C^k(D)$. Let $x \in D, u\in \mathbb{R}^n, u=(u_1,...,u_n)$ such that $ x+tu \in D, \forall t \in [0,1]$, then exits $\theta \in [0,1]$ : $$f(x+u)=f(x)+f'(x)(u)+...+\frac{1}{(k-1)!}f^{(k-1)}(x)(u)^{k-1}+\frac{1}{k!}f^{(k)}(x+\theta u)(u)^{k}\quad (1)$$ $$\lim_{u\rightarrow 0} \phi (u)=\lim_{u\rightarrow 0}{\frac{1}{k!}[f^{(k)}(x+\theta u)(u)^{k}-f^{(k)}(x)]}=0 \quad (2) $$
I can't understand why we must have the condition $f^{(k)}$ is continuos on D ($f \in C^k(D)$) instead of having $f$ is $k^{th}$-differential, because we just need to calculate $f^{(k)}(x), f^{(k)}(x+ \theta u)$ in (1)?
If now we assume that $f \in C^{k-1} (D) $ and f is $k^{th}$-differential in $D$, I think that (1) is still right, but (2) is not. Is it right? I'm try to find an example to illustrate the point, but it's quite hard for me.