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MATH
[1251]

Definitions from Global extrema

All definitions from Global extrema.

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Closed and bounded region

A region D⊆R2D\subseteq\mathbb{R}^2 is closed if it contains its boundary ∂D\partial D and bounded if it fits inside a sufficiently large disk: there is an R>0R>0 such that ∥x∥≤R\|\mathbf{x}\|\le R for every x∈D\mathbf{x}\in D.

An interior point has a neighbourhood contained entirely in DD. A boundary point has every neighbourhood meeting both DD and its complement; the set of boundary points is written ∂D\partial D.

The closed unit disk

{(x,y):x2+y2≤1}\{(x,y):x^2+y^2\le1\}

contains its boundary, the unit circle x2+y2=1x^2+y^2=1. An open disk omits that circle and is not closed.

  1. [1]:The multivariable extreme-value theorem

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Closed and bounded region