Curious function of Otto Frisch

If (x/sub n/) is a countable set of points everywhere dense on (0,1), and (j/sub n/), a set of corresponding positive numbers with ..sigma../sub 1//sup infinity/ j/sub n/ = J < infinity, then a monotone nondecreasing function f(x) on (0,1), with f(0) = 0, f(1) = J, which is continuous on (0,1) except at each x/sub n/, where it is right continuous, but left discontinuous with jump j/sub n/, is necessarily a sum of step functions uniquely determined by the x/sub n/, j/sub n/. A curious function, explicitly defined by O. Frisch, is proved to be such a function, with jump j(p/q) = 1/(2/sup q/ - 1) at each (reduced) rational point p/q on (0,1), and continuous elsewhere. Moreover, it is rational valued iff x is rational, and has some remarkable number theoretic properties, stemming from its character as a sum of step functions.