  In this paper we solve the following problem in the affirmative: Let $Z$ be a
continuum in the plane $\complex$ and suppose that $h:Z\times [0,1]\to\complex$
is an isotopy starting at the identity. Can $h$ be extended to an isotopy of
the plane?
  We will provide a new characterization of an accessible point in a planar
continuum $Z$ and use it to show that an accessible point is preserved during
the isotopy. We show next that the isotopy can be extended over hyperbolic
crosscuts. The proof makes use of the notion of a metric external ray, which
mimics the notion of a conformal external ray, but is easier to control during
an isotopy.
