@asimeqi@mathelirium What ? Continuity is an essential property of polynomials. It’s just that you can bypass these convoluted methods entirely by showing that continuity is preserverd by algebraic operations
@BraneRunner@mathelirium Yeah honestly doing that once or twice for functions just a bit more complicated than monomials can be indtructive, but more than that is a waste of time
@aapakari@Math_files It’s just as hard to come up with great definitions as it is to prove theorems. Taste for definitions is part of why Grothendieck is considered a greater mathematician than, say, Erdös, despite the latter being renowned as a great problem-solver and proof-provider
@MohamadHM3lem@MoonfinXoXo@statvora France has existed in a somexhat united form for about 1000 years. It’s much older than most countries, only a few countries like China, Japan and Iran are arguably older as continuous states
@GilGandel@DanielBenjamin8@LongFormMath In a completely different setting, and with another well-defined meaning of « infinity », the Riemann rearrangement theorem basically shows that infinity-infinity can be equal to whatever you want. So it’s not well-defined.
@GilGandel@DanielBenjamin8@LongFormMath And you have an infinity left, but that doesn’t prove infinity-infinity always equals infinity. Even simpler: take the natural numbers. Take away all of the natural numbers. You’re left with the empty set which has cardinality zero (finite).
@GilGandel@DanielBenjamin8@LongFormMath So even in the set-theoretic sense of infinity, the naive « infinity-infinity » isn’t well defined. But there’s another sense in which this fails.
@2oovy@johnald42 It’s trivial when you’ve seen this idea a thousand times (coercive function like in the fundamental theorem of algebra), but it’s not THAT trivial for a complete beginner in topology.