Lower semicontinuity’s misremembrance

2024-08-03, post № 289

mathematics, #analysis, #alternate-definition, #gastropod, #symmetry

Snails are surprisingly fast: you look at them and they creep so slowly as to strain your attention. Yet you look away for an undefined moment and their progress in distance appears wholly bewildering. Then again, I would wager to outrun most snails in a 200-meter dash.

I think symmetry’s ubiquity is an equally fickle observation; you may only be able to bear whitness to it once you have already manifested it for yourself. Notably notationally; among the following lines

$\liminf_{y\to x\neq y}f(y)\leq f(x) \quad\textbf{(Q)}$

$f(x)\leq\liminf_{y\to x\neq y}f(y) \quad\textbf{(C)}$

for a map $f:X\to\mathbb{R}$ from a topological space $X$ to the real line and a fixed point $x\in X$ , precisely one is the definition of lower semicontinuity, the other but a figment of a symmetry-ravenous mind.

I knew for a long time about $\textbf{(Q)}$ ’s questionability, but only two days ago I realized how separate a statement it is. Only the obvious implication $\lnot\textbf{(C)}\implies\textbf{(Q)}$ holds with all three other conjunctions being possible: $\textbf{(C)}\land\lnot\textbf{(Q)}$ satisfied by $1-\chi_{\{0\}}$ on $X=\mathbb{R}$ at $x=0$ , $\textbf{(C)}\land\textbf{(Q)}$ satisfied by $\chi_{]{}0,\infty{}[}$ on $X=\mathbb{R}$ at $x=0$ and $\lnot\textbf{(C)}\land\textbf{(Q)}$ satisfied by $\chi_{[0,\infty{}[}$ on $X=\mathbb{R}$ at $x=0$ .

Curiously, the infamous $\chi_{\mathbb{Q}}$ satisifies $\textbf{(Q)}$ at every point, intuitively showing $\textbf{(Q)}$ ’s inaptitude of representing an (even weakened) interpretation of continuity.