1) Take any nondecreasing function from the reals to the reals that is constant in neighborhoods of rationals, and restrict it to irrationals.  This can be constructed as a uniform limit by starting with f(0)(x) = x, enumerating the rationals as r(i) and for each i, setting f(i+1)(x) to be a sum of f(i)(x) and some piecewise linear function supported in a neighborhood of radius 2^(-i) around r(i).