Splitting One Uniform Into Two Independent Uniforms

You have a single random variable $U \sim \text{Unif}(0,1)$. Can you find measurable functions $f, g : [0,1] \to [0,1]$ such that $f(U)$ and $g(U)$ are **independent** and each is $\text{Unif}(0,1)$, without using any additional source of randomness? Provide a rigorous proof of your answer. If the answer is no, then additionally give an explicit construction of $f$ and $g$ such that $f(U) \sim \text{Unif}(0,1)$ and $g(U) \sim \text{Unif}(0,1)$ (not necessarily independent), and quantify the dependence between them -- for example, compute $\text{Cov}(f(U), g(U))$ for your construction.