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 ${\beta \geq \sqrt3]{2}}$ . It is proved that T _f is topologically transitive if inf f???????? and ${f(0)\geq\frac{1}{\beta+1}}$ . Counterexamples are provided if the assumptions are not satisfied. For ${\sqrt3]{2}\leq\beta < \sqrt{2}}$ and 0????????? 2 ? ?? it is shown that ??x?+??? (mod 1) is topologically transitive if and only if ${\alpha < \frac{1}{\beta^2+\beta}}$ or ${\alpha >2 -\beta-\frac{1}{\beta^2+\beta}}$ .