Derivative of a Real-Valued Function of Several Variables: Junghenn Defn 9.1.3

[Math Amateur]

I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...

I am currently focused on Chapter 9: "Differentiation on $\mathbb{R}^n$"

I need some help with an aspect of Definition 9.1.3 ...

Definition 9.1.3 and the relevant accompanying text read as follows:
https://www.physicsforums.com/attachments/7865
View attachment 7866

At the top of the above text, in Definition 9.1.3 we read the following text:

" ... ... there exists a vector \(\displaystyle f'(a)\) in \(\displaystyle \mathbb{R}^n\) ... ... "My question is as follows:

How (arguing from the definition of derivative) do we indicate\demonstrate\prove that \(\displaystyle f'(a) \in \mathbb{R}^n\) ... ...?Hope someone can help ... ...

Peter

 

[Ackbach]

You don't have to prove it. This is a definition (no proofs allowed). The definition is saying that if there's a vector $f'(\mathbf{a})$ satisfying the definition, then we call it the derivative. The definition is not actually claiming that the vector exists (it doesn't always, depending on $f$).

 

[Math Amateur]

You don't have to prove it. This is a definition (no proofs allowed). The definition is saying that if there's a vector $f'(\mathbf{a})$ satisfying the definition, then we call it the derivative. The definition is not actually claiming that the vector exists (it doesn't always, depending on $f$).

Thanks Ackbach ...

Appreciate the help...

Peter