Let f:R.rarw.R be a continuous function on the real line R, and denote the n-th iterate of f by f$^{n}$ :f$^{1}$=f and f$^{n}$ =f.f$^{n-1}$ for n>1. A point x.mem.R is a periodic point of f of period k>0 if f$^{k}$ (x)=x but f$^{i}$ (x).neq.x for all 0<i<k. In the recent year the following question has aroused interest : If f has a point of period k, must f also have points of other periods m.neq.kulcorner The obvious answer would seem to be "no": why should there by any connection between points of period k and points of period mulcorner Yet a little thought will show that there should be at least some results along these lines. For instance, if a continuous function f has a periodic point of period k>1, then it must also have a fixed point, by the intermediate Theorem. Also the question has an intriguing answer which was found by ths Russian mathematician Sharkovky [6] in 1964.