Are you familiar with Conway's base 13 function? It's about as badly discontinuous as a real-valued function can be.
This is fascinating. I vaguely remember this from a math class. Thanks for sharing.
That is an example of how strange real valued functions can be, and one that tests your understanding of continuity.
It's a standard example trotted out in elementary real analysis courses along with Cantor sets, etc. If you have a good lecturer, they might make you try and come up with an example like it as an exercise before you've seen it...
A good part of becoming competent at analysis is building up a grab bag of examples like this you can throw at new situations.
> In other words, it has a kink (i.e. a sharp corner) on ALL points.
x sin(1/x) is continuous and non-differentiable at 0, but I wouldn't say it has a corner there.
A sharp corner to me means a point where the left and right derivatives both exist but are different.
I don't believe this is correct. That function isn't defined at 0. While it does have a limit, continuity at a point c requires that the limit as x->c = f(c), which obviously isn't the case here.
So just define it to 0. You are correct but it's a minor detail.
Re:Calculus Is Deeply Irrational
by OscarCunningham on Wednesday August 21, 2019 @02:26AM
(#20754964)
Can a function have that property at every point?
Looks like black metal band logo. Brutal function.
You might also like the Cantor function, which is also known as "The Devils Staircase"
It is similarly used as an example, for a function that is continuous but not absolutely continuous.
https://en.wikipedia.org/wiki/Weierstrass_function