If $f:\mathbb{R}\to\mathbb{R}$ is continuous, does there exists a nonzero continuous function $g:\mathbb{R}\to\mathbb{R}$ s.t. $g(x)f(x)$ is monotone?

Is the following proposition true or false?

If $f:\mathbb{R}\to\mathbb{R}$ is continuous, then there exists a not-everywhere-zero (i.e. $g\not\equiv 0$)continuous function $g:\mathbb{R}\to\mathbb{R},\ $ such that $g(x)f(x)$ ismonotonic.

I was thinking that $f(x)=$The Weierstrass function, could be a counter-example, but I am not sure of it.