Let $U\subseteq\mathbb{R}^m$ be open and $f: U\to\mathbb{R}^n$
$f$ is differentiable in $x\in U$, if holds:
It exists $A:\mathbb{R}^m\to\mathbb{R}^n$ linear and $\varphi:\mathbb{R}^m\to\mathbb{R}^n$ with $f(x+\xi)=f(x)+A\xi+\varphi(\xi)$ for every $\xi$ in a neighborhood of $0$ with
$\lim_{\xi\to 0\\ \xi\neq 0} \frac{1}{\|\xi\|_2}\varphi(\xi)=0$
Question:
As a note it is stated, that it is sufficient that $\varphi$ is defined in a neighborhood of $0$.
I am acutally not sure why, or what this is supposed to mean. What would be an (non-trivial) example for a function, which is not defined in a neighborhood of $0$?
Of course $f:\mathbb{R}\to\mathbb{R}, x\mapsto 1/x$ is not defined in $0$, and constructing a function, which is not defined in $B_\epsilon(0)$ for $\epsilon>0$ should not be difficult, but I am actually confused.
Can you explain this side note? Thanks in advance.
The point of such definitions is merely to say that the error $$f(x+\xi) - [f(x)+A\xi]$$ is small compared to $\|\xi\|_2$ for $\xi$ sufficiently close to $0$. By definition, $\phi(\xi)$ is just this error.
If you want to try to understand it further, consider a function like $$f(x,y) = \begin{cases} 0, & x=0 \\ y, & x\ne 0\end{cases}.$$ If $f$ is differentiable at $0$, what is $A$ and does the error meet this criterion?