Lower rational approximations and Farey staircases

For a real number $x$, call $\frac{1}{n}\lfloor nx\rfloor$ the $n$-th lower rational approximation of $x$. We study the functions defined by taking the cumulative average of the first $n$ lower rational approximations of $x$, which we call the Farey staircase functions. This sequence of functions is monotonically increasing. We determine limit behavior of these functions and show that they exhibit fractal structure under appropriate normalization.