Continuity and Open Sets (Work in Progress)
From Stewart and Tall Complex Analysis 2nd Ed. Copyright 2018
Proposition 2.7. 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]Figure 2.6.[/b] Definition of continuity when [math]S[/math] is open in [math]\mathbb{C}[/math] .
[b]Corollary 2.8.[/b] If [math]S\subset\mathbb{C}[/math] is open, then [math]f:S\longrightarrow\mathbb{C}[/math] is continuous on [math]S[/math] if and only if, for every open set [math]U[/math] , the inverse image [math]f^{-1}\left(U\right)[/math] is open.](https://www.geogebra.org/resource/pG7H7vEv/Wvu4SOojBGtlZ8Wa/material-pG7H7vEv.png)
