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Is there a parametrized family of diverse continuous functions whose input and output are reals in [0.0, 1.0]?

The closest I can think of is the Beta distribution:
1) CDF works, but all the functions are monotonically increasing, so not super interesting. I can add a "flip" parameter to also get the monotonically decreasing versions.
2) PDF output is not [0.0, 1.0], but I could divide by max. Again would need to add a "flip" parameter to get the horizontally flipped versions. I could add another flag to pick between these two.

Thanks!

Alex Ravsky
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Alexei Andreev
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1 Answers1

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The set of all continuous functions from $[0,1]$ to $[0,1]$ is a semigroup $S$ with respect to a composition. The group $H(1)$ of its invertible elements is the group of homeomorphisms of $[0,1]$. In particular, $H(1)$ contains a parametrized semigroup $P=\{x^t:t>0\}$. If you want a parametrized family consisting of non-monotonic map, you can pick any non-monotonic map $g\in S$ (for instance, $g(x)=4x(1-x)$ or $g(x)=\sin(\pi x)$) and consider a parametrized family $Pg$ or $gP$). Moreover, for any parametrized family $F$ of functions from $[0,1]$ to $\Bbb R$ and a function $f$ from $\Bbb R$ to $[0,1]$ (for instance, $f(x)=\sin^2 x$ or $f(x)=\frac {2}{\pi}\arctan (x)$), $f(F)$ is a parametrized family of functions from $[0,1]$ to $[0,1]$.

Alex Ravsky
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    Thanks for the reply! This is not quite what I'm looking for, although the math is a bit above my head. I need a diverse family, so just picking ()=4(1−) is not sufficient. In theory I could manually pick a large number of such functions, but I'm trying to find a clever way to automate this.
    Another way to think about my question is in terms of kernels (like in image processing). Different kernels highlight different features. I'm doing something similar here, although not for image processing, and in 1D space.     – Alexei Andreev May 05 '20 at 17:47