Continuous implies Bounded

Continuous implies Bounded

In this video, I show that any continuous function from a closed and bounded interval to the real numbers must be bounded. The proof is very neat and involves a straightforward application of the Bolzano-Weierstraß Theorem, enjoy!

[chessematics]

He explains Analysis stuff in such a natural way that it seems like we're doing High School Algebra

[fahrenheit]

4:45 Gotta love it when you get to use the Bolzano-Weierstrass-(ouch!) theorem in a proof!

[curiosityzero]

I like these analysis videos because I'm not good with it and they help me a lot. Ty Dr Peyam

[rikthecuber]

You know what I love about your videos? You make good level math available on yt w/o the fear of views. People switch generally to simple calc1 for views.

[orenfivel]

delta Dirac impulse in 4:46, δ(t-4:46)

[punditgi]

Brilliant! Only Dr Peyam can explain these topics so well! 😊

[ashishKjr]

I learnt it this way for the first time but I think Heine Borel (equivalent form of Bolzano- Weierstrass) makes this fact intuitive. Cover [0, 1] with balls centered at each x in [0, 1] with radii 1 (say). Continuity gives a bound for f on each ball. Heine Borel gives a finite subcover and the rest is easy to see. :)

[iguana]

This proof explanation contained several insightful nuggets which helped me to understand it,

[LelPop]

Thank you so much for making my first year math course so much easier

[garvitgupta]

thank you so much for such an easy explanation, it really helped a lot, thank you so much again

[timduncankobebryant]

If only dr peyam can make a series on analysis 😕

[DrWeselcouch]

Nice video! Brings me back to my analysis days! Just wondering, what courses are you teaching in the Fall?

[desperatewanderer]

Why couldn't my teacher be like him

[taccat]

Sequences and boundedness, reminds me of Bolzano-Weierstrass

[mingyudu]

Hi will you make videos for math140B? That would be so helpful!

[cloudhuang]

Excuse me sir, isn't there posibble that f(x0) goes to infinity either? Then there is no contradiction!!??

[justincallan]

Wait so if the mother sequence goes to infinity that forces the subsequence of that mother sequence to go to infinity?

[chessematics]

2:22 and is still continuous according to the Definition.

[whatitmeans]

Is this fact just a consequence of the Extreme Value Theorem?... also, since Wienner Process is continuous, that this means that for any compact domain the path will never run to infinity, so it distribution within the compact domain cannot be gaussian (since +/- infinity is never achievable), is this right??

[pawpatrol]

what if f is only continuous at a single point?