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# 1 Scalar Functions & Limits

## Scalar Functions

Definition: A function consists of 3 things:
1. A set $X$ called the domain.
2. A set $Y$ called the co-domain.
3. A "rule" $F$ which given any $x\in X$ produces $F(x)\in Y$. We write this as $F: X\rightarrow Y$.
Definition: A scalar function, $F: \mathbb{R}^n\rightarrow\mathbb{R}$ is a function with domain $X\subseteq\mathbb{R}^n$ and co-domain $Y\subseteq\mathbb{R}$.
e.g. $f(x,y)=16-x^2-4y^2$: Domain$=\mathbb{R}^2$, Co-domain$=\{p\in\mathbb{R} | p\leq 16\}$.
Definition: The level curves of $f(x,y)$ are the curves in $\mathbb{R}^2$ of the form $f(x,y)=k$ where $k\in\mathbb{R}$. e.g. $16-x^2-4y^2=k\rightarrow x^2+4y^2=16-k$, which is an ellipse.
Definition: The cross sections of $f(x,y)$ are the curves in $\mathbb{R}^2$ given by $z=f(c,y)$ or $z=f(x,d)$ where $c,d\in\mathbb{R}$.
Tip: If $f(x,y)=g(x)+h(y)$, use cross-sections.

## Limits

Definition: The limit, L, of $f(x)$ as $x$ approaches $a$ is defined as:
$$\forall\epsilon >0\text{, }\exists\delta > 0\text{ such that if }0<|x-a|<\delta\text{, then }|f(x)-L|<\epsilon$$
When $f$ is univariate, there are only 2 ways to approach the limit, i.e. from the left and right.
Given $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and $\vec{a}\in\mathbb{R}^2$, let $\vec{x}=(x,y)$ and $\vec{a}=(a,b)$, then the norm of $\vec{x}$ and $\vec{a}$ is $$||\vec{x}-\vec{a}||=\sqrt{(x-a)^2+(y-b)^2}$$ This is the distance between $\vec{x}$ and $\vec{a}$.