Connected polynomials and continuity

As it is well-known, f∈RR is continuous if and only if f maps continua (compact, connected sets) to continua. The same holds for mappings between any two (real or complex) normed spaces. However, when we restrict ourselves to polynomials P:E→K, where E is a K-normed space, then it was proved in 2012 that P is continuous if and only if it transforms compact sets into compact sets. Here we show that (if K=C) P is continuous if and only if it transforms connected sets into connected sets. Although we provide some partial results for K=R, the general case in the real setting remains still open.