Suppose that I have a function in $f \in L^1(\mathbb{R})$ such that
$$\int\_{\mathbb{R}}f(x)v'(x)\,dx = 0$$
for all test functions $v$ which are smooth with compact support. Can I show that $f(x)$ is almost surely a constant?

This is clearly true if $f$ is smooth, but what if I just assume that $f \in L^1(\mathbb{R})$?