3 ms·
I'm in the opposite camp. Most of the calculus foundation actually depends on continuity. And one thing that has changed recently: mobile phones and zooming. E
by cyberax 2mo ago
I'm in the opposite camp. Most of the calculus foundation actually depends on continuity.
And one thing that has changed recently: mobile phones and zooming. Every child now has an almost instinctive understanding of being able to find the correct zoom level on a map so that some feature can fit completely on the screen.
And now the key insight: you can zoom-in on a continuous function indefinitely.
So the epsilon-delta formulation for the limit of functions becomes almost trivially easy to explain. And you can build from there.
I understand that the notion of continuity in itself requires limits to define it properly. But you can do that _later_. I'm speaking from experience of helping a friend's child understand calculus.
- rramadass 2mo agoWhile technology can and should be used to develop intuition, it does not help much when trying to map intuition to formalism. And that is where the epsilon-delta definition is hard to grasp; continuous is always more subtle/difficult to understand than discrete. This has already been well studied/discussed in mathematics education; Some thoughts about epsilon and delta (this is a pretty good AMS article on the subtleties involved) - https://blogs.ams.org/matheducation/2019/08/19/some-thoughts-about-epsilon-and-delta/ https://blogs.ams.org/matheducation/2019/08/19/some-thoughts... Teaching limits of sequences before limits of functions in Calculus? - https://matheducators.stackexchange.com/questions/3952/teaching-limits-of-sequences-before-limits-of-functions-in-calculus https://matheducators.stackexchange.com/questions/3952/teach... Advantages of the sequence definition of limits - https://mathoverflow.net/questions/105920/advantages-of-the-sequence-definition-of-limits https://mathoverflow.net/questions/105920/advantages-of-the-...