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 $\beta=0$.

  • $\begingroup$ Are you aware of bump functions or the intermediary functions used to build them? $\endgroup$ – Robert Wolfe Jan 29 at 14:57
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    $\begingroup$ 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. $\endgroup$ – MaoWao Jan 29 at 15:06

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