Definition
Let $A\subseteq\Bbb R^m$; let $f:A\rightarrow\Bbb R^n$. Suppose $A$ contains a neighborhood of $a$. Given $u\in\Bbb R^m$ with $u\neq 0$, define
$$
f'(a;u):=\lim_{t\rightarrow 0}\frac{f(a+tu)-f(a)}t
$$
provided the limit exists. This limit depends both on $a$ and on $u$; it is called the directional derivative of $f$ at $a$ with respect to the vector $u$.
So I would like to discuss about the hypothesis for which $A$ contains a neighborhood of $a$ that is I ask if this neighborhood must be a neighborhood in $\Bbb R^m$ or rather in relative topology of the subspace $A$. So could someone help me, please?
Best Answer
It is supposed to be an open ball surrounding $a$. The vector $u$ might be in any direction, so the idea is that for $t$ small enough $a+tu$ is still within $A$ where $f$ can be applied.
I think you are asking if the "neighborhood" is an open set in the topology of $\mathbb{R}^n$, or if it can be an open set in the topology of $A$ that comes from intersection with the topology of $\mathbb{R}^n$. But if it were the latter, why bother "supposing" $A$ contains a neighborhood of $a$? That would be the case always. So it's the former, and again, the idea behind that is as in the above paragraph.