Real Analysis | The continuous image of a compact set.

We look at some topological implications of continuity. In particular, we prove that the continuous image of a compact set of real numbers is compact and use this to prove the extreme value theorem.

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[dzakytamir]

Please make a video about the difference between continious & uniformly continious, also poinwise convergent & uniformly convergence.

[accountname]

You've already reached parity with 3b1b in my eyes mate. Nice videos

[beatoriche]

Not to be rude or anything, but there’s a bit of notation used in this video that I think is not exactly optimal from a didactic point of view. A little pet peeve of mine is when people use the same notation for images under a mapping and the value of the mapping at a single point, i. e. writing _f(x)_ for an element _x_ of the domain and _f(G)_ for a given subset _G_ of the domain. I think it’s didactically unfortunate that so many math textbooks denote these very different things identically because it adds to the confusion many students already have with regards to set theory. Many first-year or second-year math students struggle quite a bit with sets and, among other things, with the difference between an element of a set and a subset of a set. I think math notation in general – though I’m primarily concerned with material directed at math students in their first few years – should clearly distinguish these concepts for this exact reason. To show a specific example of why this notation is bad, consider a function _f_ with the set _{1, {1}}_ as its domain. What exactly does _f({1})_ mean here? Is it the value of _f_ at the point _{1}_, or is it the image of the set _{1}_ – a proper subset of the domain – under _f_? Instead denoting image sets as _f[G]_ would establish a clear distinction; _f({1})_ would then unambiguously refer to the value of the function at the element _{1}_ of its domain, and _f[{1}]_ would be the image set of the set _{1}_. This may seem like a bit of an obscure example to some readers, but nothing could be further from the truth! You may not encounter these types of things in a real analysis course, but in various more advanced settings – such as in measure theory or topology – sets, set families, and power sets are thrown around left and right, and in these cases, this conflation ends up being extremely confusing. I remember being really intimidated by this stuff when I first picked up a topology textbook, and I don’t think I’m alone with this. Of course, I’m well-aware that the notation I’m criticizing is so exceedingly common that I’m definitely on a sinking ship here; however, I wanted to put this out there as a bit of food for thought.

[toygun]

guys and Sir Penn, would you recommend topology master programme? Is it too difficult or enjoyable to content like pure math?

[Flammewar]

Nice, I was just trying to prove this 3 hours ago. :D

[Unidentifying]

@Caladbolg, thats so weird man just as you replied I figured it out and deleted my comment, then saw the notification of your reply afterwards, sorry about that one -- thank you so much though, very kind of you <3!!! YEs i removed that second loop and it worked!

[RolandThePaladin]

A simpler counterexample for question 1 is taking B to be R and f to be any constant function.

[tomkerruish]

Had an idea this was coming (in stark contrast with absolutely everything you've said about vertex algebras.) Next up: continuous functions on compact domains are uniformly continuous?

[TheMauror]

What a cool video, I love these series! Could someone recommend me some analysis books so I could learn some more?

[fanrco]

Can someone explain to me why the Extreme Value theorem even exists? Is it not just saying "the biggest and lowest values in a set are in the set"? isn't that self evident just by definition?

[JalebJay]

I'm here before a good place to stop.