Timeline for Existence of function satisfying $f(f'(x))=x$ almost everywhere
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 5, 2019 at 21:52 | comment | added | Paul | Peter is right. the question is not ye solved | |
| Aug 5, 2019 at 7:41 | comment | added | Pietro Majer | There is a solution of the form $\alpha(-x)^\beta$ for $x<0$, but with another $\alpha$ and $\beta$, and it has $\beta<0$, so this solution is unbounded near $0$. | |
| Aug 5, 2019 at 6:41 | comment | added | Pietro Majer | Almost, @Marc Chamberland. Your function is odd, its derivative is even, so the composition is even, and we get $|x|$ again. | |
| Aug 4, 2019 at 19:37 | vote | accept | Paul | ||
| Aug 5, 2019 at 21:53 | |||||
| Aug 4, 2019 at 19:14 | comment | added | Marc Chamberland | Almost, @FusRoDah 1 . The function $f(x)=a|x|^b$ is even, so $f(f^\prime(x))=|x|$. Instead, choose $f(x)=ax^b$ for $x>0$ and $f(x)=-a(-x)^b$ if $x<0$. | |
| Aug 4, 2019 at 19:13 | comment | added | Pietro Majer | @FusRoDah are you sure? Your function has constant sign, while x can be both positive and negative | |
| Aug 4, 2019 at 18:07 | comment | added | FusRoDah | But if we take $f(x)=a|x|^b$, then it will be differentiable everywhere except $0$, and the condition will be satisfied for all $\mathbb R$. So $f(x)=a|x|^b$ with $a$, $b$ as above is a solution to the OP's problem. | |
| Aug 4, 2019 at 14:53 | comment | added | Paul | But we must have $f∘f′(x)=x$ a.e $x\in \mathbb{R}$ ( for x<0, f(x)=?) If the question is to Study the existence of a continuous function $f : \mathbb{R_+^*} \rightarrow \mathbb{R_+^*}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R_+^*}$ The answer is yes, there is even a $C^1$ function $f : \mathbb{R_+^*} \rightarrow \mathbb{R_+^*}$ satisfying $\forall x\in\mathbb{R_+^*},\mbox{ } f\circ f'(x)=x.$ we can take $f(x)=ax^b,\forall x>0$ with $b=(1+\sqrt 5)/2$ and $a=\exp(\frac{-b\ln b}{b+1})$ | |
| Aug 4, 2019 at 14:13 | history | answered | Marc Chamberland | CC BY-SA 4.0 |