4 ms·
If you want to appreciate how strange real numbers are, check out Weierstrass function. It's a function that is continuous everywhere, but differentiable nowher
by gnulinux 7y ago
If you want to appreciate how strange real numbers are, check out Weierstrass function. It's a function that is continuous everywhere, but differentiable nowhere. In other words, it has a kink (i.e. a sharp corner) on ALL points.
https://en.wikipedia.org/wiki/Weierstrass_function https://en.wikipedia.org/wiki/Weierstrass_function
- ska 7y agoThat 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.
- proc0 7y agoLooks like black metal band logo. Brutal function.
- ska 7y agoYou 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. cfL https://en.wikipedia.org/wiki/Cantor_function https://en.wikipedia.org/wiki/Cantor_function
- lanstin 7y agoIf you like these sorts of things, there is a slim tome called "Counterexamples in Topology" which is very long list of similar weird things.
- joker3 7y agoAre you familiar with Conway's base 13 function? It's about as badly discontinuous as a real-valued function can be. https://en.wikipedia.org/wiki/Conway_base_13_function https://en.wikipedia.org/wiki/Conway_base_13_function
- gnulinux 7y agoThis is fascinating. I vaguely remember this from a math class. Thanks for sharing.
- im3w1l 7y ago> 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.
- OscarCunningham 7y agoCan a function have that property at every point?
- im3w1l 7y agoNo. https://math.stackexchange.com/questions/1342427/if-a-function-is-left-and-right-differentiable-everywhere-how-much-can-the-one https://math.stackexchange.com/questions/1342427/if-a-functi...
- wutbrodo 7y agoI 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.
- im3w1l 7y agoSo just define it to 0. You are correct but it's a minor detail.