Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

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 lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.