For those who are interested in this topic, you can refer to "Introduction to real analysis" by Bartle and Sherbet, 4th edition. The continuous extension thm is given on pages 144, 145. Sir, is it possible to extend this result to higher dimensions? (like the boundary of an open set.)
If memory serves, Hunt used projections onto Walsh distributions to finally pin down the convergence properties of Fourier transforms in the early 60s (or maybe 70s). Maybe you could make a special exception for them. (ps Hunt was a great hand ball player, but often Phelps was better; however, Hunt could understand the "corner walking" vibration functions for certain degenerate ball shots into the corners.)
I'm not sure I understand how an endpoint can be continuous. Can you have continuity on only one side? In other words, in the first example, can we say ~f is continuous at x=0 because lim_a->0+ (~f(a)) = ~f(0) ? Or is it only continuous because x sin(1/x) is continuous on all point of the interval (-1,0) as well?
Hello sir, thank you very much for the videos they are very very helpful for clearing the concept, but I have a doubt in this last minutes of the video you showed that if x -> a then (f^~)(x) -> (f^~)(a), I completely understand the part where you showed that the limit exists and is unique irrespective of the sequence we choose, but I have doubt that how can we be sure that the limit say S obtained is same as (f^~)(a), I.e. S = (f^~)(a)......???? Waiting for you reply.. Thank you😇😇
Hello Dr.Peyam! I want to ask about the oscillating sequences: what is the mathematical importance of the central value of oscillation? Should we have to know its value? Thanks
It's not hard to show that sin 1/x is not UC, Let delta be 2. For any epsilon, let k be the ceiling of 1/epsilon. Then let x be 2/π(4k+1) and y be 2/π(4k+3). |x-y| < epsilon and yet |f(x)-f(y)|=delta.
おおおー
ちょうど連続写像とかの話を読んでたから、めっちゃ参考になるわぁああ
ありがたいこっちゃでほんと😌🙏🙏🙏
For those who are interested in this topic, you can refer to "Introduction to real analysis" by Bartle and Sherbet, 4th edition. The continuous extension thm is given on pages 144, 145.
Sir, is it possible to extend this result to higher dimensions? (like the boundary of an open set.)
Yes. And also on the complex domain.
If memory serves, Hunt used projections onto Walsh distributions to finally pin down the convergence properties of Fourier transforms in the early 60s (or maybe 70s). Maybe you could make a special exception for them. (ps Hunt was a great hand ball player, but often Phelps was better; however, Hunt could understand the "corner walking" vibration functions for certain degenerate ball shots into the corners.)
Greetings Doc!, you explain very well, and your english is very clear!
I prefer f "snek" for the tilde but "squiggle" works as well I guess
Very nice explanation,sir. Thank you very much!
I don't know much on calc 3(except the very basics) but can you suggest me a video on cross product of 2 vectors w/o converting into components?
I'm not sure I understand how an endpoint can be continuous. Can you have continuity on only one side?
In other words, in the first example, can we say ~f is continuous at x=0 because
lim_a->0+ (~f(a)) = ~f(0) ?
Or is it only continuous because x sin(1/x) is continuous on all point of the interval (-1,0) as well?
Yes, it’s called one sided continuity, so here f is continuous at 0 from the right
Hello sir, thank you very much for the videos they are very very helpful for clearing the concept, but I have a doubt in this last minutes of the video you showed that if x -> a then (f^~)(x) -> (f^~)(a), I completely understand the part where you showed that the limit exists and is unique irrespective of the sequence we choose, but I have doubt that how can we be sure that the limit say S obtained is same as (f^~)(a), I.e. S = (f^~)(a)......???? Waiting for you reply..
Thank you😇😇
Hello Dr.Peyam!
I want to ask about the oscillating sequences:
what is the mathematical importance of the central value of oscillation?
Should we have to know its value?
Thanks
Glad I subscribed! ✌️🤝
I expected the proof to use reliance on a Limit existing AND all derivatives existing, as you approach the end of the domain segment.
Doctor , do you want to tell us new definition for cont ?
Or definition cont on set bigger than last set
Wow I just had this last week in topology class
~f(0)=0
~f(x)=sin(1/x), x∈(0,1)
~f(x)=sin(1)-(x-1)cos(1), x∈[1,∞)
Also, what about analytic continuation?
We redefine the function on the extended set to get the extended connection
Is correct doctor?
Yes.
Very good lecture. Thanks Dr 3.14159....m .
Here is an interesting question I came across ' how many Spheres of 1 cm radius will fit inside larger hollow sphere of 1meter'.
It's not hard to show that sin 1/x is not UC, Let delta be 2. For any epsilon, let k be the ceiling of 1/epsilon. Then let x be 2/π(4k+1) and y be 2/π(4k+3). |x-y| < epsilon and yet |f(x)-f(y)|=delta.
Not evident. Thank you very much.
Tune in for the math, stay for the dad jokes :-)
Sometimes life is just sinx/x near zero
Hahaha
@@drpeyam باور کن
Wow the Doc does not accept extensions, that's harsh haha
I was wondering how Oreo is doing?
Oreo passed away in June 😞
Sorry for your loss :(
I think this video is just an excuse to say squiggle 😃😄