continue

Def 16.1

$f$ is continue imply
$d(x,x^{\prime })<\epsilon ⟹d(f(x),f(x^{\prime }))<\epsilon$

Thm 16.2

$f$ is continue at x $⟺$ $({\lim }_n{x}_n=x⟹{\lim }_{n}f(x_n)=f(x))$

Thm 16.3 17.2

$f:x⟶y$ TFAE

1. $f$ is continuous
2. it’s inverse image $f^{-1}(O)$ is open for all open image $O$

Thm 17.5

A Metric Space $(X,d)$ and $f,g\in C(X,\mathbb{R})$

1. $f+g\in C(X,\mathbb{R})$
2. $f\times g\in C(X,\mathbb{R})$
3. $\exists \varphi :\mathbb{R}⟶\mathbb{R}\text{ c.t. }⟹\varphi \circ f\in C(X,\mathbb{R})$
   where $C$ is the set of all continue functions from $X$ to $\mathbb{R}$

Thm 17.7

The uniform limit of continuous functions is continue

Thm

continue $⟺$ sequentially continue

Thm 35.2

inverse function of continue function is continue.