  In this paper we study the density in the real line of oscillating sequences
of the form
  $$ (g(k)\cdot F(k\alpha))_{k \in \mathbb{N}} ,$$ where $g$ is a positive
increasing function and $F$ a real continuous 1-periodic function. This extends
work by Berend, Boshernitzan and Kolesnik who established differential
properties on the function $F$ ensuring that the oscillating sequence is dense
modulo $1$.
  More precisely, when $F$ has finitely many roots in $[0,1)$, we provide
necessary and also sufficient conditions for the oscillating sequence under
consideration to be dense in $\mathbb{R}$. All the results are stated in terms
of the Diophantine properties of $\alpha$, with the help of the theory of
continued fractions.
