If $\beta$ is a continuous function with $\beta f'=0$ for all $f\in C_c^\infty$, are we able to conclude $\beta=0$?

Question

Let $\beta:\mathbb R\to\mathbb R$ be continuous. If $$\beta f'=0\;\;\;\text{for all }f\in C_c^\infty(\mathbb R),\tag1$$ are we able to conclude $\beta=0$?

If, given a compact $K\subseteq\mathbb R$, we could find an $f\in C_c^\infty(\mathbb R)$ with $$K\subseteq\operatorname{supp}f\tag2,$$ $(1)$ would imply that $\beta$ vanishes on all compact subsets of $\mathbb R$, which (by continuity) clearly would imply $\beta=0$.