A nondegenerate metric space
that is both compact and connected is called a continuum. In this paper it is proved
that if M is a continuum with the property that for each indecomposable
subcontinuum H of M there is a continuum K in M containing H such that K is
connected im kleinen at some point of H and if f is a continuous function on M into
the plane, then the boundary of each complementary domain of f(M) is hereditarily
decomposable. Consequently, if M is a continuum in Euclidean n-space that does not
contain an indecomposable continuum in its boundary, then no planar continuous
image of M has an indecomposable continuum in the boundary of one of its
complementary domains.
