Let us say $X$ is a random variable with cdf $F$. I know that when $F$ is invertible then $F(X)$ has unif$(0,1)$ distribution. The proof goes like $$P[U\le u] = P[F(X)\le u] = P[X\le F^{-1}(u)] = F(F^{-1}(u))=u$$ The proof does not go through when $F$ is not invertible. What can we say in general? Can we say something if $F$ is continuous?
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