- Take any nondecreasing continuous 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).
S. Carnahan ♦
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