Prop. 2.12, Cor. 2.13, & Figure 2.6: Continuity and Open Sets (Work in Progress)
From Stewart and Tall Complex Analysis 2nd Ed. Copyright 2018
Proposition 2.12. A complex function is continuous if and only if, for every
open set in , the set is open in .
Proof. Suppose that is continuous and is open. Let ). Then so
there exists such that . By continuity of there exists such
that
Hence
and is open. Conversely, suppose that is open in for every open set . Given and , the set is open, so is open in and there exists such that
Hence
so is continuous. EOP.![[b]Corollary 2.13.[/b] Is [math]S[/math] is open, then [math]f[/math] is continuous if and only if, for every
open set [math]U[/math] , the inverse image [math]f^{-1}\left(U\right)[/math] is open.
[b]
Figure 2.6.[/b] Definition of continuity when [math]S[/math] is open in [math]\mathbb{C}[/math] .](https://www.geogebra.org/resource/bk9hFS4G/JLCVUdAz3x5HMbUc/material-bk9hFS4G.png)
