  In this paper we prove a characterization of continuity for polynomials on a
normed space. Namely, we prove that a polynomial is continuous if and only if
it maps compact sets into compact sets. We also provide a partial answer to the
question as to whether a polynomial is continuous if and only if it transforms
connected sets into connected sets. These results motivate the natural question
as to how many non-continuous polynomials there are on an infinite dimensional
normed space. A problem on the \emph{lineability} of the sets of non-continuous
polynomials and multilinear mappings on infinite dimensional normed spaces is
answered.
