Topological transitivity for a class of monotonic mod one transformations

Suppose that f : [0, 1] → [0, 2] is a continuous strictly increasing piecewise differentiable function, and define T f x := f ( x ) (mod 1). Let . It is proved that T f is topologically transitive if inf f ′ ≥ β and . Counterexamples are provided if the assumptions are not satisfied. For and 0 ≤ α ≤ 2 − β it is shown that βx + α (mod 1) is topologically transitive if and only if or .