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

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 $$\beta=0$$.

• Are you aware of bump functions or the intermediary functions used to build them? – Robert Wolfe Jan 29 at 14:57
• All you really need is that for all $x\in\mathbb{R}$ there exists $f\in C_c^\infty(\mathbb{R})$ such that $f'(x)\neq 0$. Since you can translate test functions, this boils down to showing the existence of one non-trivial test function, which is extremely standard (see e.g. en.wikipedia.org/wiki/Bump_function). That being said, there are test functions that satisfy (2). You can construct them by mollifying the indiactor function by convolution with a bump function. All of this has certainly been discussed many times on this site. – MaoWao Jan 29 at 15:06