Is the sequence of functions $f_n=\sin(nx)$ equicontinous in $\mathbb{R}$?

Question

I've proved that the family $f_n=\sin(nx)$ is not equicontinous on the Interval $[0,1]$. We can prove also that the sequence is not equicontinous on any compact Interval. But I am wondering : Can we generalize that the family is not equicontinous for the complete $\mathbb{R}$ ?

Answer 1

This is the opposite of a generalization. If the family is not equicontinous on some set, then that family of functions will not be equicontinous when extended to some superset either.