For a scheme $X$, the sheaf $\mu_n$ is defined by $\mu_n(U) = $ $n$-th roots of unity in $\mathscr O_X(U)$. Assuming that $n$ is coptime to the characteristic of $X$, $\mu_n$ is certainly locally constant in the etale topology.
However, I doubt that $\mu_n$ is locally constant in the Zariski topology. On the other hand, I am having trouble finding examples of $X$ where $\mu_n$ is not locally constant.
So are there such examples? Or is $\mu_n$ always locally constant even for the Zariski topology?
Edit: I found a counter example almost immediately after posting the question. However, I would still be interested in more counterexamples, in particular I am interested in whether such examples can occur where $X$ is a scheme over a field $k$.
Other classes of counter examples are also welcome.
Edit 2: I found a counter example over a field using the same idea as before(see posted answer) . Now I am interested in a genuinely non affine example.
Consider $R = \mathbb Z[1/n,2\zeta_n]$ where $\zeta_n$ is a primitive n-th root of unity and suppose $2$ is coprime to $n$. Then $R$ does not contain $\zeta_n$ but $R[1/2]$ does contain $\zeta_n$. Since $R$ is constant, this shows that $\mu_n$ is not locally constant.
Take $R = k[x, y] /(x^n - y^n) $ where k did not contain $\zeta_n$. On inverting $x$, we get a root of unity.