Linear operator exists then differentiable?

Question

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such that

$$\lim\limits_{\vec{h} \rightarrow 0} \frac{||\vec{f}(\vec{x_0}+\vec{h})-\vec{f}(\vec{x_0})-A\vec{h}||}{||\vec{h}||} = 0$$

Then, by definition, $\vec{f}$ is differentiable, and $A =: d\vec{f}(\vec{x_0})$.

I'm wondering what this definition is called and where can I read more about this?

Thanks

Answer 1

It is not necessary to put arrows on top. These lecture notes by Terrence Tao explain the concept quite well.

Answer 2

It is called Frechet derivative.