[Analysis] Update to new helper import scheme, continue differentiability summary

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex

@@ -0,0 +1,14 @@
+\newsectionNoPB
+\subsection{The differential}
+\setLabelNumber{all}{2}
+\compactdef{Differentiable function} We have function $f: X \rightarrow \R^m$, linear map $u : \R^n \rightarrow \R^m$ and $x_0 \in X$. $f$ is differentiable at $x_0$ with differential $u$ if
+$\displaystyle \lim_{\elementstack{x \rightarrow x_0}{x \neq x_0}} \frac{1}{||x - x_0||} (f(x) - f(x_0) - u(x - x_0) = 0$ where the limit is in $\R^m$.
+We denote $\dx f(x_0) = u$.
+If $f$ is differentiable at every $x_0 \in X$, then $f$ is differentiable on $X$
+
+% ────────────────────────────────────────────────────────────────────
+\stepLabelNumber{all}
+\shortproposition Let $f, g : X \rightarrow \R^m$ with $X \subseteq \R^n$ open
+\begin{itemize}[noitemsep]
+    \item The function $f + g$ is differentiable with differential $\dx (f + g) = \dx f + \dx g$
+\end{itemize}