Hello !

I want to prove thatthe supremum of continuous functions is a lower semi continuous function .Could someone help me please ?I tried to find on the internet something to help me ,but i had no luck!

Printable View

- Mar 19th 2011, 05:45 AMasteriaslower semi continuous function
Hello !

I want to prove that**the supremum of continuous functions is a lower semi continuous function .**Could someone help me please ?I tried to find on the internet something to help me ,but i had no luck!

- Mar 19th 2011, 11:04 AMFernandoRevilla
The result is also valid for the supremum $\displaystyle f=\sup \{f_i:\;i\in I\}$ of lower semicontinuous functions $\displaystyle f_i:X\to \mathhbb{R}$ , $\displaystyle (X,T)$ topological space .

:__Hint__

$\displaystyle G_i=f_i^{-1}(\alpha,+\infty)\in T$ for all $\displaystyle i\in I$ so, $\displaystyle G=\bigcup\limit_{i\in I}G_i \in T$

. - Apr 3rd 2011, 03:09 AMasterias
Thank you very much for your help :-)