Timeline for properties of orderd upper and lower semi continuous functions [closed]

10 events

when toggle format	what		by	license	comment
Dec 13, 2016 at 13:59	history	edited	Xifeng Su	CC BY-SA 3.0	added 230 characters in body; edited title
Dec 13, 2016 at 13:41	history	edited	Xifeng Su	CC BY-SA 3.0	added 5 characters in body
Dec 1, 2016 at 13:05	history	closed	Pietro Majer Wolfgang Alexey Ustinov Alex Degtyarev Stefan Kohl♦		Not suitable for this site
Dec 1, 2016 at 6:54	review	Close votes
Dec 1, 2016 at 13:05
Nov 30, 2016 at 22:00	comment	added	Pietro Majer		It is true that for any $f$ LSC and $g$ USC on $X$, with $f\ge g$, there is a continuous $h$ in between. It's Katetov's insertion theorem, actually a characterization of normal spaces $X$.
Nov 30, 2016 at 21:56	comment	added	Pietro Majer		Xifeng Su: no, not even if you assume all $f_n$ and $g_n$ to be continuous: take $X:=[0,1]$, $g_n(x)=\min( nx_+,1)$, $f_n(x)=\min( (nx+1)_+,1)$.
Nov 30, 2016 at 20:56	answer	added	Alexandre Eremenko		timeline score: 1
Nov 30, 2016 at 19:00	comment	added	Xifeng Su		Thank Fedor Petrov! What about $\{f_n\}_{n\geq1}$ is lower semi continuous, decreasing and $\{g_n\}_{n\geq1}$ is upper semi-continuous, increasing and $f_n(x)\geq g_n(x)$? Is it true to construct a continuous function $h(x)$ in between?
Nov 30, 2016 at 15:00	comment	added	Fedor Petrov		What if $X=[-1,1]$, $f_n$ are all equal to the characteristic function of $[0,1]$, $g_n$ to that of $(0,1]$?
Nov 30, 2016 at 12:00	history	asked	Xifeng Su	CC BY-SA 3.0