Timeline for Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 28, 2021 at 16:13 | comment | added | Dave L Renfro | @Timothy Chow (and others who might be interested): For a lot about the type of nondifferentiability behavior that "most" continuous functions have, see my answer to Generic Elements of a Set. | |
| Jul 28, 2021 at 2:20 | comment | added | Timothy Chow | @Stef See the math.SE question, 'Amount' of nowhere-differentiable functions in C([0,1])?. | |
| Jul 27, 2021 at 18:42 | comment | added | Dave L Renfro | @Stef: There are $2^{2^{\aleph_0}}$ many nowhere differentiable functions, even $2^{2^{\aleph_0}}$ many functions having no points of continuity. For example, each of the $2^{2^{\aleph_0}}$ many functions in $\{f \cup g: \; f\in [0,1]^{\mathbb Q} \; \text{and} \; g\in [2,3]^{{\mathbb R} - {\mathbb Q}}\,\}$ is continuous at no point. | |
| Jul 27, 2021 at 16:16 | comment | added | Buzz | @Stef Continuous functions originating from the same point have a relatively natural measure on them, in the form of Brownian motion (Wiener measure). Under that measure, almost all the continuous functions are nowhere differentiable. | |
| Jul 27, 2021 at 15:32 | comment | added | Stef | How many continuous real functions are nowhere differentiable? | |
| Jul 27, 2021 at 13:56 | comment | added | David E Speyer | Moreover, there are $2^{2^{\aleph_0}}$ functions from $\mathbb{R}$ to $\mathbb{R}$, but only $2^{\aleph_0}$ of them are continuous. (Because a continuous function is determined by its values on the rational numbers.) So, in a very simple sense, almost all functions are discontinuous. | |
| S Jul 27, 2021 at 3:59 | history | answered | Orntt | CC BY-SA 4.0 | |
| S Jul 27, 2021 at 3:59 | history | made wiki | Post Made Community Wiki by Orntt |