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Thread moved from the technical forums, so no Homework Template is shown

Let x denote the position of a particle on the number line. From x, it can move to either the point a-a

It’s fairly clear the particle can get arbitrarily close to the origin (by moving in one direction continuously and then suddenly swapping). Thus you can get arbitrarily close to a point that is reached by moving in one direction continuously. But I don’t know how to fill in the “holes” in between these points

^{2}+ax or to the point x-ax-a+a^{2}for some fixed 0<a<1. Suppose the particle starts at the origin. Prove that any open interval that is a subset of the interval (a-1,a) contains a point that the particle can reach.It’s fairly clear the particle can get arbitrarily close to the origin (by moving in one direction continuously and then suddenly swapping). Thus you can get arbitrarily close to a point that is reached by moving in one direction continuously. But I don’t know how to fill in the “holes” in between these points