  However, we have so far not used another continuity assumption which is also a reasonable assumption. So far we have only made the following demand: the state the dial is in at stage S2 must be a continuous function of the state of the dial at stage S3. But, the state of the dial at stage S2 is arrived at by taking the state of the dial at stage S1, and rotating it over some angle. It is not merely the case that the effect of the interaction, namely the state of the dial at stage S2, should be a continuous function of the cause, namely the state of the dial at stage S3. It is additionally the case that path taken to get there, the way the dial is rotated between stages S1 and S2 must be a continuous function of the state at stage S3. And, rather surprisingly, it turns out that this can not be done. Let us illustrate what the problem is before going to a more general demonstration that there must be a fixed point solution in the dial case.