I always found the topological definition of continuity to be very intuitive and really much simpler than the classical ε-δ definition, not to mention it is much more widely applicable. This video got me thinking about the question of whether there are any real analysis courses that consistently use the usual topological definition of continuity from the beginning. I'm guessing it's not very practical because all the topology will have to be translated back into the language of real numbers anyway, but it might be more intuitive for first-time learners that continuous functions are basically just deformations and that if you consider any open or closed set in the output space, it can only have been produced by an open or closed set, respectively.
Ah, a function is continuous if, when I draw it, there are no filled dots required. Doesn't matter if I have to pick up the pen, as long as I'm crossing between the open intervals where the function is defined.
Ok. Thank you very much. Why demonstrate that the definition coincide vith metric space in the part two ? Why the second part is not sufficient ? And moreover why not the first part only ?
But why does this need to be there I agree that not all topological spaces have the notion of a metric or distance still...... Is there any case where the ε-δ definition fails? I don't think so. Then why was there a necessity of the topological version of continuity was it discovered by accident and not out of necessity or just discovered? Bruh I am confused somebody help
I love your videos' thumbnail designs: plain, simple and eye-catching, none too clustered.
Thank you!!! They take hours to make
I always found the topological definition of continuity to be very intuitive and really much simpler than the classical ε-δ definition, not to mention it is much more widely applicable. This video got me thinking about the question of whether there are any real analysis courses that consistently use the usual topological definition of continuity from the beginning. I'm guessing it's not very practical because all the topology will have to be translated back into the language of real numbers anyway, but it might be more intuitive for first-time learners that continuous functions are basically just deformations and that if you consider any open or closed set in the output space, it can only have been produced by an open or closed set, respectively.
This is a must watch for every students in calculus/analysis
Hi Dr. Peyam,
I’m a calc2 student right now, and I love your videos/method of presentation. Thank you!!
Thanks so much!!!
Ah, a function is continuous if, when I draw it, there are no filled dots required. Doesn't matter if I have to pick up the pen, as long as I'm crossing between the open intervals where the function is defined.
love your videos!
Ok. Thank you very much.
Why demonstrate that the definition coincide vith metric space in the part two ? Why the second part is not sufficient ? And moreover why not the first part only ?
But why does this need to be there I agree that not all topological spaces have the notion of a metric or distance still......
Is there any case where the ε-δ definition fails? I don't think so. Then why was there a necessity of the topological version of continuity was it discovered by accident and not out of necessity or just discovered?
Bruh I am confused somebody help
Profesor regresé please ✨ se lo extraña en UA-cam 😢
Dr Peyam, you look and gesticulate like Sheldon Cooper
@drPeyam, please respond to me. Are you the famous actor in Sheldon Cooper?
Brillant!
So so helpful!
Glad I could help :)
It's very nice. I'm don't speak english, but, i understand perfectly...
Thank you, I like your explanation
How
Leoj Agumbay My explanations are very good 😉
Woah
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3:48 peyam machine broke
Not again!!! LOL 😂