Timeline for Generalisation of 'derivatives are Baire 1'
Current License: CC BY-SA 4.0
4 events
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| Jun 15, 2023 at 7:08 | comment | added | Sam Sanders | @Gro-Tsen Thanks for the answer, which is helpful: I am also interested in the behaviour of functions you mention ($f'=g$ ae), i.e. the what happens on the measure zero set is not that important. | |
| Jun 14, 2023 at 19:55 | comment | added | Gro-Tsen | (The difference between the result in my previous comment and your question is that you impose $g(x)=0$ when $f'(x)$ does not exist, whereas the above result simply requires $f' = g$ almost everywhere. But I still think it's interesting enough to mention.) | |
| Jun 14, 2023 at 19:54 | comment | added | Gro-Tsen | Not an answer but still worth noting is the following result: A. Bruckner, Differentiation of Real Functions (2d ed, 1994), chapter 8 (“Derivatives a.e. and generalizations”), theorem 1.3 (rephrased to align with your notation): a necessary and sufficient condition on a function $g\colon [a,b]\to\mathbb{R}$ for there to exists a function $f$ such that $f' = g$ almost everywhere is that $f$ be measurable and finite almost everywhere; furthermore, when this condition is satisfied, $f$ can be chosen continuous. | |
| Jun 14, 2023 at 19:28 | history | asked | Sam Sanders | CC BY-SA 4.0 |