  One of the main questions in the field of complex dynamics is the question
whether the Mandelbrot set is locally connected, and related to this, for which
maps the Julia set is locally connected. In this paper we shall prove the
following
  Main Theorem: Let $f$ be a polynomial of the form $f(z)=z^d +c$ with $d$ an
even integer and $c$ real. Then the Julia set of $f$ is either totally
disconnected or locally connected. In particular, the Julia set of $z^2+c$ is
locally connected if $c \in [-2,1/4]$ and totally disconnected otherwise.
