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In this paper we study the density in the real line of oscillating sequencesof the form  $$ (g(k)\cdot F(k\alpha))_{k \in \mathbb{N}} ,$$ where $g$ is a positiveincreasing function and $F$ a real continuous 1-periodic function. This extendswork by Berend, Boshernitzan and Kolesnik who established differentialproperties on the function $F$ ensuring that the oscillating sequence is densemodulo $1$.  More precisely, when $F$ has finitely many roots in $[0,1)$, we providenecessary and also sufficient conditions for the oscillating sequence underconsideration to be dense in $\mathbb{R}$. All the results are stated in termsof the Diophantine properties of $\alpha$, with the help of the theory ofcontinued fractions.

Density of Oscillating Sequences in the Real Line