\begin{proof} See Lewis pg. 56-57 \end{proof}

Now, if a map $A$ exists satisfying $(1)$, the idea is that the map $\boldsymbol x\mapsto f(\boldsymbol x_{0})-A(\boldsymbol x-\boldsymbol x_{0})$ provides a linear approximation for $f$ at $\boldsymbol x_{0}$.

It is now convenient to introduce notation describing approximations of various forms. >[!def] Landau symbols >1. When we write $o(\lVert \boldsymbol x \rVert_{\mathbb{R}^{n}}^{k})$ we mean a continuous map from a neighbourhood of $\mathbf{0}\in \mathbb{R}^{n}$ to $\mathbb{R}^{m}$ that satisfies $$\lim_{ \boldsymbol x \to \mathbf{0} } \frac{o(\lVert \boldsymbol x \rVert _{\mathbb{R}^{n}}^{k})}{\lVert \boldsymbol x \rVert_{\mathbb{R}^{n}}^{k} }=0$$ >2. We shall also use the notation $O(\lVert \boldsymbol x \rVert_{\mathbb{R}^{n}}^{k})$ which stands for a map defined in the Neighbourhood of $\mathbf{0}\in \mathbb{R}^{n}$ taking values in $\mathbb{R}^{m}$, and satisfying $$\lVert O(\lVert \boldsymbol x \rVert _{\mathbb{R}^{n}}^{k}) \rVert _{\mathbb{R}^{m}}\le M\lVert \boldsymbol x \rVert ^{k}_{\mathbb{R}^{n}}$$for some $M>0$.

With this notation says that there exists at most one $A\in \mathscr{L}(\mathbb{R}^{n};\mathbb{R}^{m})$ such that$$f(\boldsymbol x)=f(\boldsymbol x_{0})+A(\boldsymbol x-\boldsymbol x_{0})+o(\lVert \boldsymbol x-\boldsymbol x_{0} \rVert _{\mathbb{R}^{n}})$$ We can now define the derivative and differentiability