I've been playing around with composing translations with the modulus function.
Let $f_0(x) = |x|$, and $f_{n+1}(x) = \left| f_n(x)-\frac{1}{n+1}\right|$, where $n$ is a positive integer.
I would like to understand $\displaystyle{\lim_{n \to \infty} f_n}$, but I have no idea where to start.
The graphs $y=f_k(x)$ start to look like fractals, even for small $k$. Here are the graphs for $4 \le k \le 7$. The final picture is $y=f_{20}(x)$, but the line in too thick to show the detail.
