Aug 24 Balls and Open Sets

Balls

MathJax TeX Test Page
As we went over in the limit definition post, whenever x is within $\delta$ of p, f(x) is within $\epsilon$ of $f(p)$. Balls are generalizations of this. A ball of radius $\delta$ centered at $p$ is defined by
$$B(p, \delta) = \{x \in \mathbb{R}^n \mid ||x-p|| < \delta\}$$
So, this is a higher dimensional analog of a delta-range or an epsilon-range.

Open Sets

MathJax TeX Test Page
What is an open set? Just think of it as a region with no boundary. That means for every point, there exists a tiny region around that point that's still in the region.
For example, if a point is 1 unit from the "boundary", then all points 0.5 units from the point are inside the region. That gives us our mathematically accurate definition.
$$\text{A subset U}\subseteq \mathbb{R}^n \text{ is open means } \forall p \in U, \exists \delta \gt 0 s.t. B(p, \delta) \subseteq U$$