For me, the proof only works, or works most clearly, if f is monotonic. Otherwise, I see the a an b sequences getting stuck: a can’t move towards b when f(c) is +, and b can’t move towards a when f(c) is -. But it’s not a problem overall, because interval I can be the result of the process applied to sub intervals.
@@brightsideofmaths Didn’t know that. I thought c was just an arbitrary x. BTW, my son is majoring in Math at the University of Michigan in Ann Arbor. I am studying your course because I like puzzles, and Real Analysis has a reputation for being challenging. I am now up to video 32, having watched most multiple times. I absolutely love the rigor. And the approaches are so clever, a glimpse of genius often.
@@brightsideofmaths I’m not sure why the definition of continuity doesn’t prove intermediate values. P.S. I have an amazing proof of Euler’s Basel Problem I’d like to share, the most naturally/easiest (heuristic ) I’ve ever seen; others feel contrived to me. How can I show it to you?
![Real Analysis 33 | Some Continuous Functions [dark version]](http://i.ytimg.com/vi/TS6ngvHdMs8/mqdefault.jpg)
![Real Analysis 33 | Some Continuous Functions [dark version]](/img/tr.png)
![Real Analysis 40 | Local Extreme and Rolle's Theorem [dark version]](http://i.ytimg.com/vi/Sh-NfW9MdVI/mqdefault.jpg)
i totally support your efforts and hard work ,
For me, the proof only works, or works most clearly, if f is monotonic. Otherwise, I see the a an b sequences getting stuck: a can’t move towards b when f(c) is +, and b can’t move towards a when f(c) is -.
But it’s not a problem overall, because interval I can be the result of the process applied to sub intervals.
But c is always the middle point. So we will always move :)
@@brightsideofmaths Didn’t know that. I thought c was just an arbitrary x. BTW, my son is majoring in Math at the University of Michigan in Ann Arbor.
I am studying your course because I like puzzles, and Real Analysis has a reputation for being challenging.
I am now up to video 32, having watched most multiple times. I absolutely love the rigor. And the approaches are so clever, a glimpse of genius often.
Really nice! Thanks :)
@@brightsideofmaths I’m not sure why the definition of continuity doesn’t prove intermediate values.
P.S. I have an amazing proof of Euler’s Basel Problem I’d like to share, the most naturally/easiest (heuristic ) I’ve ever seen; others feel contrived to me.
How can I show it to you?
@@212ntruesdale I am not an expert there. So maybe someone else :)
very good explanation , but i already know this very well
Glad to hear that :D