# 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

for a map from a topological space to the real line and a fixed point , 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 ’s questionability, but only two days ago I realized how separate a statement it is. Only the obvious implication holds with all three other conjunctions being possible: satisfied by on at , satisfied by on at and satisfied by on at .

Curiously, the infamous satisifies at every point, intuitively showing ’s inaptitude of representing an (even weakened) interpretation of continuity.