(Solved): Problem \( 1(10+10=20 \) points \( ) \) Give an example of each of the following, or argue that suc ...

Problem \( 1(10+10=20 \) points \( ) \) Give an example of each of the following, or argue that such a request is impossible. (a) A function \( f: \mathbb{R} \rightarrow \mathbb{R} \) which is nowhere differentiable, but for which \( |f| \) is differentiable everywhere. (b) A continuous function \( f:[0,1] \rightarrow \mathbb{R} \) such that \( f \) is not constant and \( f(x)=f\left(x^{2}\right) \) for all \( x \in[0,1] \).