Open Sets, Limits, and Continuity

Open Sets

The open disk $D_r({\mathbf{x}}_0)$ of radius $r$ and center ${\mathbf{x}}_0$ is defined to be the set of all points $\mathbf{x}$ such that $\Vert \mathbf{x}-{\mathbf{x}}_0\Vert <r$.

Then, an open set $U\subset {\mathbb{R}}^n$ is one such that for all ${\mathbf{x}}_0\in U$ there exists some $r>0$ such that $D_r({\mathbf{x}}_0)\subset U$. Intuitively, a set is open when the “boundary points” are not included. $\varnothing$ is open.

Example: $A=\left\{(x,y)\in {\mathbb{R}}^2\right\}|y<0$ is an open set. Intuitively, no matter what point you take from $A$ with distance $l$ from the $y$-axis, you can “fit” an open disc $D_{0<r<l}$.

Note that $\stackrel{\sim }{A}=\left\{(x,y)\in {\mathbb{R}}^2\right\}|y\le 0$ is not an open set. If you take any point where $y=0$, you could not fit any open disc.

Theorem 1: For each ${\mathbf{x}}_0\in {\mathbb{R}}^n$ and $r>0$, $D_r({\mathbf{x}}_0)$ is an open set.

A neighborhood of $\mathbf{x}\in {\mathbb{R}}^n$ means an open set $U$ containing $\mathbf{x}$.

$\mathbf{x}\in {\mathbb{R}}^n$ is a boundary point of $A\subset {\mathbb{R}}^n$ if every neighborhood of $\mathbf{x}$ contains at least one point in $A$ and at least one point not in $A$. $\mathbf{x}$ need not be in $A$. The boundary points of an open set are not contained.

Limits and Continuity

A sequence is a set of numbers which you can enumerate. Of course, the integers are a sequence, along with any function of the integers. The set of rational numbers is also a sequence (it takes some cool steps to see this).

A sequence $x_1,x_2,\dots$ converges to $x_0$ if for any radius $r>0$ there exists $N(r)$ such that for all $n>N,x_n\in D_r(x_n)$.

Definition: Suppose $f:A\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$, the limit $\underset{x\to x_0}{\lim }f(x)=b$ if for any sequence $x_1,x_2,\dots$ which converges to $x_0$, $f(x_1),f(x_2),\dots$ should converge to $b$

$\underset{x\to x_0}{\lim }f(x)=f(x_0)$

Example: $f:\mathbb{R}\to \mathbb{R}$
$f(x)=\{\begin{array}{l}1\text{ if }x>0\\ -1\text{ if }x\le 0\end{array}$
$\underset{x\to x_0}{\lim }f(x)=\{\begin{array}{l}1\text{ if }{x}_0>0\\ -1\text{ if }{x}_0<0\\ ?\text{ if }x=0\end{array}$
The definition of a limit specifies that it must converge to the same value for any sequence that converges to $x_0$, so the regular limit is not defined

However, there are valid one-sided limits $\underset{x\to x_0^{-}}{\lim }$ and $\underset{x\to x_0^{+}}{\lim }$ (in higher dimensions there is a notion of directional limits)

Sometimes simplification is necessary to find an limit to an undefined point
$\underset{x\to 1}{\lim }\frac{x-1}{\sqrt{x}-1}=\underset{x\to 1}{\lim }\frac{(x-1)(\sqrt{x}+1)}{x-1}=2$

Definition: A function $f:A\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$ is continuous at $x_0\in A$ if $\underset{x\to x_0}{\lim }f(x)=f(x_0)$, and $f$ is a continuous function if it is continuous for all $x_0\in A$

Example: Polynomials $p(x)=a_0+a_{1}x+\cdots +a_n{x}^n$ are continuous

Some properties of continuity (these correspond to limit laws),

1. $f⟹cf$
2. $f_1,f_2,\dots ,f_n⟹f_1+f_2+\cdots +f_n$
3. $f,g⟹fg$
4. $f\ne 0⟹\frac{1}{f}$
5. $f=(f_1,f_2,\dots ,f_m)⟺f_1,f_2,\dots ,f_m$
6. $f,g⟹g\circ f$