Is there a continuous function that accepts a single uniform random variable and returns two independent uniform random variables?

I can define a function $f(X) = (Y_1,Y_2)$ that accepts a random variable $X$ with a uniform distribution on $[0,1]$, and returns two independent uniform random variables $Y_1,Y_2$. This function generates $Y_1$ by taking every even digit of $X$ and generates $Y_2$ by taking every odd digit of $X$.

Is there a continuous function that can accept a random variable $X$ that has a uniform distribution on $[0,1]$ and returns two independent uniform random variables $Y_1,Y_2$?

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Source : Link , Question Author : gigalord , Answer Author : Community

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