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Definition: Neighborhoods

A neighborhood of a point $\b{x} \in \R^n $ is the set

$$\set{ \b{y} \in \R^n : \norm{ \b{y} - \b{x}} < \epsilon} $$

where $\epsilon $ is some positive number. The neighborhood is also called the ball with the radius $\epsilon $ and center $\b{x} $.

Definition: Interior Points

A point $\b{x} \in S$ is said to be an interior point of the set $S$ if the set $S$ contains some neighborhood of $\b{x}$, that is, if all points within some neighborhood of $\b{x} $ are also in $S $.

The set of all the interior points of $S$ is called the interior of $S $.

Definition: Boundary Points

A point $\b{x} $ is said to be a boundary point of the set $S $ if every neighborhood of $\b{x} $ contains a point in $S $ and a point not in $S $.

Note that a boundary point of $S $ may or may not be an element of $S $. The set of all boundary points of $S $ is called the boundary of $S $.

Definition: Open Sets and Closed Sets

A set $S $ is said to be open if it contains a neighborhood of each of its points, that is, if each of its points is an interior point, or equivalently, if $S $ contains no boundary points.

A set $S $ is said to be closed if it contains its boundary. We can show that a set is closed if and only if its complement is open.

Theorem : Theorem of Weierstrass

$f: \Omega \to \R $, $f $ is a continuous function, $\Omega \subseteq \R^n $ is a compact set.

$$\exists \b{x}_0 \in \Omega\text{ such that } \forall x \in \Omega, f(\b{x} _0) \leq f(\b{x}). $$

In other words, $f $ achieves its minimum on $\Omega $.