I am completely stuck to solve Problem 7.2 Case 2) in the book A Probabilistic Theory of Pattern Recognition, Devroye et al.
The problem asks to generalize Theorem 7.1 and 7.2 about slow rates of convergence when $X$ is uniformly distributed on $[0,1)$ and the regression function $\eta$ is infinitely continuously differentiable.
However, by the following reasoning I show a contradiction. Theorem 7.1 assumes that the Bayes risk is equal to $0$ therefore $\eta(X) = 0$ or $1$ $\mathbb{P}_X$-almost surely. Since $\eta$ is continuous, it implies that $\eta = 0$ or $1$ on $[0,1)$. Therefore $Y=1$ or $0$ almost-surely independently of $X$ and Theorem 7.1 for example can not hold.
Can you help me to find my mistake and give me a hint to solve this problem ?