A discussion about the definition of derivative.

Question

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?

2'5 9'2 · Accepted Answer · 2020-09-18T20:48:00.493

2

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.

edited Sep 18 '20 at 20:48

answered Sep 18 '20 at 20:44

2'5 9'2

56,991

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I'm saying ${a}\subset \text{some open ball}\subset A \subset \mathbb{R}^n$. And the ball is a full, open ball, not just the intersection of some open ball with $A$. – 2'5 9'2 Sep 18 '20 at 20:49
So it has not sense to calculate the derivative in the boundary points? – Antonio Maria Di Mauro Sep 18 '20 at 20:54
So substantially I ask how to explain what Munkres state here. – Antonio Maria Di Mauro Sep 18 '20 at 21:13
@AntonioMariaDiMauro You could calculate some directional derivatives in boundary points. Not the full derivative. – Henno Brandsma Sep 18 '20 at 22:37
@HennoBrandsma Okay, so what it is wrote here is probably a typo? What can you say about? – Antonio Maria Di Mauro Sep 19 '20 at 07:43
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@AntonioMariaDiMauro The boundary of half space is an exception, we have all directions, but only from side $t>0$, so we take limits from the right in $\Bbb R$ to define the derivative there. – Henno Brandsma Sep 19 '20 at 08:05

A discussion about the definition of derivative.

1 Answers1