  For a real number $x$, call $\frac1n \lfloor nx \rfloor$ 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.
