The assumption of continuity is important. Without this assumption, the discontinuous piecewise linear function f : [ 0 , 3 ) → → [ 0 , 3 ) defined as: f : x ↦ ↦ { x + 1 f o r 0 ≤ ≤ x < 2 x − − 2 f o r 2 ≤ ≤ x < 3 for which every value has period 3, would be a counterexample. Similarly essential is the assumption of f being defined on an interval. Otherwise f : x ↦ ↦ ( 1 − − x ) − − 1 , which is defined on real numbers except the one: R ∖ ∖ { 1 } , and for which every non-zero value has period 3, would be a counterexample.