A Natural Interpretation of an Artificial Function

Abstract

which is intended to show that the concept of continuity is more subtle than one expects, because f is continuous exactly the irrational points (see [1, p. 86] or [2, p. 94]). And for the average student, this is the only property of the function f, that is, it is not interesting otherwise because there seems to be no reasonable context in which this function arises naturally. Its strange continuity behavior is only just punishment for having created a function by an artificial f-definition! But there is an interesting interpretation of this function: We consider only positive arguments, and we interpret t as the slope of a ray in the first quadrant of the x-y plane, starting the origin. Let us intercept the ray x = n, where n is any natural number chosen at random. Then f(t) is the that the y-coordinate of the interception point is also an integer, because this is the case if and only if n is divisible by q, and thus in making a great number of random guesses of the integer x-coordinate n we shall have also an integer y-coordinate in one of q cases. (Strictly speaking, there is no probability measure on N that makes the choice of each natural number equally probable! But if you choose your number n among the natural numbers 1, 2,. . ., N, randomly with respect to the uniform probability measure, your chance to get one which is divisible by q is [N/q]/N, which tends to l/q for N -> oo. So f(t) = l/q is the asymptotic density that n chosen in N is divisible by q.) Of course, this interpretation of the function f may be reformulated for a torus instead of the x-y plane. A slight modification of the function f doesn't change its continuity behavior but admits another nice interpretation: Place a square billiard table in the first quadrant of the x-y plane, with its lower-left corner the origin, and let a ball start this corner and proceed along the line with slope t > 0. Then