In a previous question on MO I mentioned that I had convinced myself of the following:

When $f_p(z) = z - p(z)/p'(z)$ and $p$ is a complex polynomial, the Julia set, $J(f_p)$ is not continuously determined (in the sense of the Hausdorff metric) by the roots of $p$ if any of those roots is not simple.

I can think of a few different ways for a proof to go, but the main idea is the same in each one: A multiple root can be "separated" by an arbitrarily small perturbation to form a set of simple roots, and that causes a change in the dynamics.

I have worked out the details (I think) for one proof along those lines, but per the advice I received on meta I will not ask for anyone to read my work in order to answer my question. My question is:

Does this result appear in the literature (or is it a trivial application of something that appears in the literature)?

Alternatively, if the result is actually wrong then I would of course accept that as an answer.