4.2 Limits and continuity  (Page 3/14)

Definition

Let S be a subset of ${ℝ}^2$ ( [link] ).

1. A point $P_0$ is called an interior point    of $S$ if there is a $\delta$ disk centered around $P_0$ contained completely in $S.$
2. A point $P_0$ is called a boundary point    of $S$ if every $\delta$ disk centered around $P_0$ contains points both inside and outside $S.$

Definition

Let S be a subset of ${ℝ}^2$ ( [link] ).

1. $S$ is called an open set    if every point of $S$ is an interior point.
2. $S$ is called a closed set    if it contains all its boundary points.

Definition

Let S be a subset of ${ℝ}^2$ ( [link] ).

1. An open set $S$ is a connected set    if it cannot be represented as the union of two or more disjoint, nonempty open subsets.
2. A set $S$ is a region    if it is open, connected, and nonempty.

Definition

Let $f$ be a function of two variables, $x$ and $y,$ and suppose $\left(a,b\right)$ is on the boundary of the domain of $f.$ Then, the limit of $f\left(x,y\right)$ as $\left(x,y\right)$ approaches $\left(a,b\right)$ is $L,$ written

if for any $\epsilon >0,$ there exists a number $\delta >0$ such that for any point $\left(x,y\right)$ inside the domain of $f$ and within a suitably small distance positive $\delta$ of $\left(a,b\right),$ the value of $f\left(x,y\right)$ is no more than $\epsilon$ away from $L$ ( [link] ). Using symbols, we can write: For any $\epsilon >0,$ there exists a number $\delta >0$ such that