# 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 $$Y_1,Y_2$$?