Let $U \subset \mathbb R^{d}$ be an open set, $x \in U$ and $f: U \to \mathbb R$ partially differentiable, it states:
$\partial_{k}f$ is continuous $\forall k \in \{1,...,d\} \iff f$ is differentiable in $x$, the fundamental matrix is continuous in $x$ and is equal to $Df(x)$ (we defined this as the Jacobi-Matrix).
Problem: I am struggling to understand this concept of fundamental matrix being significant, as I immediately assume that the only matrix that plays a role here is the Jacobi-Matrix. Furthermore, why is the notion of continuity in $x$ for the Jacobi-Matrix important? Isn't the Jacobi-Matrix a linear transformation $Df: \mathbb R^{d} \to \mathbb R$ and so naturally continuous. If not, under what circumstances is the matrix not continuous $x$ and how does this affect the differentiability of $f$ in $x$, seeing as though:
$f(x+h)=f(x)+Df(x)(h)+r(h)$ with $h \to 0$
I am struggling to see when a Jacobi-Matrix is not continuous.
We've just commenced this topic, so it is a bit overwhelming, but anyone who is willing to shed more light on multidimensional differentiation, is greatly appreciated.
As a linear transformation, $Df(x) \in \mathcal{L}(\mathbb{R}^d,\mathbb{R})$, the mapping $h \mapsto Df(x) (h)$ is continuous (for any $x$).
However, the mapping $x \to Df(x)$ may or may not be continuous with respect to a particular norm on $\mathcal{L}(\mathbb{R}^d,\mathbb{R})$, that is
$$\|y - x\| < \delta \implies \|Df(y) - Df(x)\|_{\mathcal{L}(\mathbb{R}^d,\mathbb{R})} < \epsilon$$
Specifically, the collection of linear transformations $\mathcal{L}(\mathbb{R}^d,\mathbb{R})$ is a normed space under the norm
$$ \|L\|_{\mathcal{L}(\mathbb{R}^d,\mathbb{R})} = \sup \{|L(h)|: \,h \in \mathbb{R}^d,\, \|h\| \leqslant 1 \},$$
and we can show that
$$\|Df(y) - Df(x)\|_{\mathcal{L}(\mathbb{R}^d,\mathbb{R})} \leqslant \left(\sum_{k=1}^d |\partial_k f(y) - \partial_k f(x)|^2\right)^{1/2} $$
Whence, continuity of the partial derivatives at $x$ implies continuity of $x \mapsto Df(x)$.