I'm reading about higher derivative from textbook Analysis I by Amann.
In this section, the author said that
Suppose now that $m \in \mathbb{N}^{\times}$ and $\partial^{m-1} f: X \rightarrow \mathcal{L}^{m-1}(E, F)$ is already defined.
In my understanding, "$\partial^{m-1} f: X \rightarrow \mathcal{L}^{m-1}(E, F)$ is already defined" means that $\partial^{m-1} f (x)$ exists for all $x \in X$. It follows from this statement that if $\partial^{m} f\left(x_{0}\right)$ exists then $\partial^{m-1} f\left(x\right)$ exists for all $x \in X$. Hence, we only define the $m$-th derivative at $x_0$ if the $(m-1)$-th derivative exists for all $x$ in the domain of $f$.
Could you please confirm if my understanding is correct? Thank you for your help!
Taking the given definition literally, what you have written is correct. However, the given definition is not standard, since to define $\partial^mf(x_0)$, you really only need $\partial^{m-1}f(x)$ to be defined for all $x$ in some neighborhood $U$ of $x_0$, since that's all you need to define the derivative of $\partial^{m-1}f$ at $x_0$. So, when most people talk about $\partial^mf(x_0)$, they do not require $\partial^{m-1}f$ to be defined everywhere, but only in a neighborhood of $x_0$.
(Very often this is a moot point because you want $\partial^mf$ to exist not just at a single point $x_0$ but at every point of $X$, in which case the two definitions are equivalent.)