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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[사기꾼진우야내가-tt]

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.

[hydraslair]

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.

[tomkerruish]

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.

[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.

[natepolidoro]

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

[vincentliu]

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

[pde]

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

[noumaneelgaou]

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

[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?

[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!

[JSSTyger]

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