Consider the functions $(f_n(x))$ defined recursively by $$f_0(x)=|x-1|$$ $$f_{n+1}(x)=|x-f_n(x)|$$ Then determine the points on $\mathbb{R}$ where the function $f_n(x)$ is differentiable.
Further suppose $$S=\{a \in \mathbb{R} : (f_n(a)) \text{ converges in } \mathbb{R} \}$$
Then to what function does the sequence of functions $(f_n(x))$ converge pointwise on $S$?
My Attempt I was able to find $f_1(x)$ easily. However the analysis becomes complicated for me for higher $n$'s. I feel that induction will help, but I cannot proceed further.