So, recently i got a bit (too) curious about approximating $x\mapsto x^{p/q}$ for arbitrary values of $p/q\in{]0,1[}$, starting with the following piecewise linear interpolation for all integer $n\ge1$ and $t\in[0,1]$ : $$\left\{\begin{array}{l} f_0:{]0,+\infty[}\to{]0,+\infty[}\\ f_0\big((n+1)^qt+n^q(1-t)\big)=(n+1)^pt+n^p(1-t)\\ \displaystyle f_0\left(\frac t{n^q}+\frac{1-t}{(n+1)^q}\right)=\frac t{n^p}+\frac{1-t}{(n+1)^p} \end{array}\right.$$ I then used a property of such functions, that is for all $x>0$, we have $x^{p/q}=1/(1/x^{p/q})$, to design a sequence that would, i'd hoped, converge to the real values of $x\mapsto x^{p/q}$, as such, for all integer $n\ge0$ : $$\begin{array}{lllll} f_{n+1}&:&]0,+\infty[&\longrightarrow&]0,+\infty[\\ &&x&\longmapsto&\displaystyle\frac12\left(f_n(x)+\frac1{f_n\left(\frac1x\right)}\right) \end{array}$$ I was mainly focusing on $p/q=1/2$ more specifically, and to my surprise it worked ! And yes, even if $p\wedge q\ne1$. It's when i tried looking at this for different values, however, that i figured out something odd ; it somehow doesn't work if $p/q\ne1/2$ ? I tried to figure out why on my own, but i've been completely unable to figure it out. This is a really random question, i know, but i wanted to know if it were a well-known phenomenon ? I saw nothing remotely close to this, i'm guessing there's something to do with derivatives and stuff but i can't exactly pinpoint what.
I mean, sure, at least i got it for $p/q=1/2$, so i can just apply it as many times as i want to get all the $1/2^k$ and by multiplication with $x\mapsto x^n$ for $n\in\mathbb N$ and by density of dyadics in $\mathbb Q$ (and additionally $\mathbb R$ then) i can get anything i want i guess, but that's not a very satisfying conclusion in my opinion, so... Yeah, i wanted to know what you all had to say about it ?
I've made a visualisation on Desmos here : https://www.desmos.com/calculator/jeaobv5zwr, though only on $[1/2^q,2^q]$ because I was a bit lazy, but it's better than nothing. The squiggly function below is just the logarithm of the substractive difference between the approximation (blue) and the actual $x\mapsto x^{p/q}$ function (red).