\subtext{Treating functions $f: X \subset \R^n \to \R / \C / \R^m,\quad m,n \geq 1$}

\notation $f(x)$ for $f: I \subset \R^n \to \R^m$ means:\\
$x = (x_1, \ldots, x_n),\quad f(x) = f\bigl( f_1(x), \ldots, f_m(x) \bigr)$

\subsection{Multivariate functions}

\definition \textbf{Linear map} $f: \R^n \to \R^m$\\
\subtext{In other words: $f(x) = \textbf{A}x,\quad \textbf{A} \in \C^{m \times n}$}

Linear Maps are continuous

\definition \textbf{Affine Linear map} $f(x) \mapsto \textbf{A}x + c$

\definition \textbf{Quadratic form} $Q: \R^n \to \R$\\
\subtext{In other words: $Q(x) = \sum_{i=0}^{n}\sum_{j=0}^{m}\left( a_{i,j}x_i x_j \right)$}

\definition \textbf{Monomials} $M(x): \R^n \to \R \mapsto \alpha x_1^{d_1}\cdots x_n^{d_n}$\\
\subtext{For example: $f(x, y, z) = 16x^2yz^5$}

\definition $\deg(M) := e = \sum_{i=1}^{n} d_i$\\
\subtext{For example: $\deg(16x^2yz^5) = 8$}

\definition \textbf{Polynomials} $P(x) := \sum_{i=0}^{n} M_i(x)$\\
\subtext{For example: $P(x,y,z) = x^3 + 25x^2y^6z + xy$}

Polynomials are continuous.

\definition $\deg(P) := d \geq \max \{ \deg(M_i) \sep M_i \text{ in } P \}$\\
\subtext{For example: $\deg(x^3 + 25x^2y^6z + xy) = 9$}

Visualisations for some function types:

\definition \textbf{Graph} $G_f := \{(x,y,z) \in \R^3 \sep z = f(x,y) \}$\\
\subtext{Only for $f: \R^2 \to \R$. Visually, this is a surface in $\R^3$}

\definition \textbf{Vector Plots} for $f: \R^2 \to \R^2$\\
\subtext{Points in $(x,y) \in \R^2$ are displayed as vectors $f(x,y)$}

\newpage
\subsection{Sequences in $\R^n$}

\definition \textbf{Sequences in $\R^n$}\\
$(x_k)_{k \geq 1}$ s.t. $x_k \in \R^n$ where $x_k = \bigl( x_{k,1},\ldots x_{k,n} \bigr)$

\definition \textbf{Convergence in $\R^n$}\\
$$
    \lim_{k \to \infty} \Bigl( x_k \Bigr) = y \iff \forall \epsilon > 0, \exists N \geq 1: \forall k \geq N:\quad || x_k - y || < \epsilon
$$

Using this definition preserves many familiar results:

\lemma \textbf{Equivalent conditions to Convergence}\\
$
\begin{array}{ll}
    (i)     & \forall i \text{ s.t. } 1 \leq i \leq n:\quad \underset{k \to \infty}{\lim} \Bigl(x_{k,i}\Bigr) = y_i \\
    (ii)    & \underset{k \to \infty}{\lim} \Big\| x_k - y \Big\| = 0 
\end{array}
$

\definition \textbf{Limits at points}
\begin{align*}
    & \underset{x \neq x_0 \to x_0}{\lim} \Bigl( f(x) \Bigr) = y \iffdef \forall \epsilon > 0, \exists \delta > 0: \\
    & \forall x \neq x_0 \in X: \big\| x - x_0 \big\| < \delta \implies \big\| f(x) - y \big\| < \epsilon
\end{align*}
\subtext{$X \subset \R^n,\quad f:X \to \R^m,\quad x_0 \in X,\quad y \in \R^m$}

The sequence test for Continuity works for point-limits too.

\subsection{Continuity in $\R^n$}

\definition \textbf{Continuity in $\R^n$}\\
$f \text{ continuous at } x_0 \in X \iffdef \forall \epsilon > 0, \exists \delta > 0:\\$
$$
    \big\| x - x_0 \big\| < \delta \implies \big\| f(x) - f(x_0) \big\| < \epsilon\\
$$
$f$ continuous $\iffdef \forall x \in X: f$ continuous at $x$\\
\subtext{$X \subset \R^n,\quad f:X \to \R^m$}

\lemma \textbf{Continuitiy using Sequences}\\
$f$ continuous at $x_0$ if and only if:
$$
    \forall (x_k)_{k \geq 1}:\quad \underset{k \to \infty}{\lim} \Bigl( x_k \Bigr) = x_0 \implies  \underset{k \to \infty}{\lim}\Bigl(f(x_k)\Bigr) = f(x_0)
$$
\subtext{$X \subset \R^n,\quad f:X \to \R^m$}

\lemma \textbf{Continuity of Compositions}\\
$f: X \to Y,\ g: Y \to \R^p \text{ continuous } \implies g \circ f  \text{ continuous}$\\
\subtext{$X \subset \R^n,\quad Y \subset \R^m,\quad p \geq 1$}

\lemma \textbf{Continuity using Coordinate Functions}\\
$f: \R^n \to \R^m$ continuous $\iff \forall i \leq m: f_i$ continuous

\subsection{Subsets of $\R^n$}

\definition \textbf{Bounded}\\
$X \subset \R^n$ bounded $\iffdef \Bigl\{ \big\| x \big\| \sep x \in X \Bigr\} \subset \R$ bounded.\\
\subtext{Example: The open disc $D = \{ x \in \R^n \sep \big\| x - x_0 \big\| < r \}$ is bounded.}

\definition \textbf{Closed}\\
$X \subset \R^n$ closed $\iffdef \forall (x_k)_{k\geq 1} \in X:\quad \underset{x \to \infty}{\lim}\Bigl( x_k \Bigr) \in X$\\
\subtext{Example: $\emptyset$, $\R^n$ are closed.}

\definition \textbf{Compact} if closed and bounded.\\
\subtext{Example: The closed Disc $\Lambda = \{ x \in \R^n \sep \big\| x - x_0 \big\| \leq r \}$ is compact.}

\definition \textbf{Open}\\
$X \subset \R^n$ open $\iffdef \forall x \in X,\ \exists \delta > 0:$
$$\bigl\{ y \in \R^n \sep  |x_i - y_i| < \delta,\ \ \forall i \leq n \bigr\} \subset X$$
\subtext{In other words: Changing any coord. $x_i$ by $\delta$ keeps $x'$ in $X$}\\
\subtext{Example: $\emptyset, \R^n$ are open (and closed)}


\lemma The Cartesian Product preserves bounded/closed.

\lemma \textbf{Continous functions preserve closed/open}

\smalltext{$\forall \text{ closed/open } Y:$}
$$
    f^{-1}(Y) = \bigl\{ x \in \R^n \sep f(x) \in Y \bigr\} \text{ is closed/open.}
$$
\subtext{$f: \R^n \to \R^m$ is continuous,$\quad Y \subset \R^m$}\\
\subtext{Note: $X$ open/closed, does \textit{not} imply $f(X)$ open/closed}

\lemma \textbf{The complement of open sets is closed}
$$
    X \subset \R^n \text{ is open } \iff \underbrace{\bigl\{ x \in \R^n \sep x \notin X \bigr\}}_{\text{Complement}} \text{ is closed}
$$

\begin{subbox}{Min-Max Theorem}
    \smalltext{For compact, non-empty $X \subset \R^n$, continuous $f: X \to \R$:}
    $$
        \exists x_1,x_2 \in X :\quad f(x_1) = \underset{x \in X}{\sup} f(x),\quad f(x_2) = \underset{x \in X}{\inf} f(x)
    $$
\end{subbox}