Sharkovskii also proved the converse theorem: every upper set of the above order is the set of periods for some continuous function from an interval to itself. In fact all such sets of periods are achieved by the family of functions T h : [ 0 , 1 ] → → [ 0 , 1 ] , x ↦ ↦ min ( h , 1 − − 2 | x − − 1 / 2 | ) for h ∈ ∈ [ 0 , 1 ] , except for the empty set of periods which is achieved by T : R → → R , x ↦ ↦ x + 1 .