Real Analysis | Continuity, connected sets, and the IVT.

We prove that the continuous image of a connected set is connected and show this result implies the intermediate value theorem.
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[@TranquilSeaOfMath]

Nice presentation.

[@goodplacetostop2973]

14:19

[@alexandergarfin422]

i enjoyed this thoroughly

[@hydraslair4723]

I really hope you can do some point-set topology and cover this theorem in a more general setting! I think it's a beautiful result especially for metric spaces.

[@user-nq1ok2ih8z]

The intermediate value theorem is equivalent to saying that every continuous image of an interval is also an interval. Being an interval is a property of a set conserved under continuous map.

[@tomkerruish2982]

Just started watching, so I don't know if he later corrects this. E and F must also be nonempty, for otherwise every set A is disconnected, by taking E = A and F = Ø.

[@err954]

Ξφηβχκ

[@tomkerruish2982]

@@err954 ???

[@vincentliu5245]

6:22 couldn't B just be the subset of C or D? and not necessarily equal?

[@heewahhin7470]

If x ∈ C ∪ D = (f^-1(E) ∩ B) ∪ (f^-1(F) ∩ B) = (f^-1(E) ∪ f^-1(F)) ∩ B Which implies x ∈ B Therefore we have B = C ∪ D

[@YUYANGHONG]

At 7:47 maybe D contains the isolated point of C, but due to the intersection of E and F is empty, this could not exist.

[@pde9322]

What about the converse part of the theorem ? Is the converse part also true? Or there is a case of strictly increasing…

[@Reliquancy]

What’s the word for sets that are disconnected just because a single point like (3,5) U (5,7)? Or finitely many such points?

[@hybmnzz2658]

Disjoint

[@Reliquancy]

ゴゴ Joji Joestar ゴゴ But disjoint could also be (3,5) U (8,11) it seems like there would be some more specific term for what I described.

[@schweinmachtbree1013]

@@Reliquancy (3,5) U (5,7) is the "punctured epsilon-neighborhood" or "deleted epsilon-neighborhood" of 5, where epsilon=2, or you might just call it a "punctured/deleted open ball centered at 5". if the "puncture" didn't not occur in the center of the set -- say we had (3,6) U (6,7) -- then you can still call it a "punctured (open) neighborhood of 6". I don't think there is a term for when we have more than one "puncture" (although one can of course write it as an intersection of punctured neighborhoods of the form I have just described).

[@Reliquancy]

schweinmachtbree Cool, thanks.

[@Reliquancy]

schweinmachtbree I was thinking about an interval of the real line where you remove all the rationals is the reason I was asking, I guess that’s even an infinite number of punctures.

[@natepolidoro4565]

The birth of algebra and calculus... just magical.

[@sayanjitb]

How one can distinguish between theorem , corollary and lemma? I know the definitions. But how to use them properly while writing in proper mathematical settings, it bothers me! Can anyone here help me out!

[@schweinmachtbree1013]

a *corollary* is an immediate (or very simple) consequence of a result (could be a theorem, proposition, lemma, or even another corollary) that has just been proved - for example in this video, the "Calculus 1 IVT" is a corollary of the "topological IVT", which is the main focus of the video. a *proposition* is a very simple result, almost always following straight from the definitions, e.g. that the composition of two injections is an injection, that the divisibility relation is transitive, or proving that something is an equivalence relation (so corollaries and propositions both have simple proofs, but the former falls out from a previous result(s), while the latter follows straight from the definitions)

[@schweinmachtbree1013]

a *lemma* is a non-trivial result that is needed to prove a "more important result", which will be a theorem (on the other hand, a trivial/easy result that is used to prove a more important result is usually a proposition). different mathematicians of course all have different styles, so for a theorem which has a big proof, some mathematicians will "take out" a part of the proof and state it as a lemma. I can think of two examples of this off hand: 1) one of the standard proofs of the "Calculus 1 extreme value theorem" begins by proving that a continuous function on a closed bounded interval is bounded, and then goes on to show that it attains its sup and inf, so some mathematicians might like to "take out" the first part and state it as "the boundedness lemma" (see the second paragraph at en.wikipedia.org/wiki/Extreme_value_theorem). 2) a special case of Tychonoff's theorem states that the product of two compact topological spaces is compact. One can give a self-contained proof this result, but the set-theoretical content is a little "hardcore", so instead one can first prove what is called the Tube Lemma (en.wikipedia.org/wiki/Tube_lemma) and then apply it to the special case of Tychonoff's theorem, which cuts down the set-theoretical parts considerably

[@schweinmachtbree1013]

and a *theorem* is anything which isn't a proposition, lemma, or corollary :) theorems tend to be substantive, and if it really substantive and/or deep we might call it a "fundamental theorem", e.g. the fundamental theorem of arithmetic, the fundamental theorem of calculus, and the fundamental theorem of algebra.

[@sayanjitb]

@@schweinmachtbree1013 i am spellbound, you have provided each nitty gritty details of my question whose answer i am trying to find out for last 1 year but no one could give this much satisfactory result. Thank you so much. NB I am thinking to take a print out of it.

[@sayanjitb]

And another thank you for providing hyper links also!

[@noumaneelgaou1624]

Hy my beautiful theater l holp you will make the video correct de problem of IMO 2020

[@arvindsrinivasan424]

🔥🔥🔥

[@JSSTyger]

We mere mortals are but CHALK and DUST....CHALK AND DUST, MAXIMUS!!

[@ernestonarvarte4605]

first

[@kiwi4731]

Nope