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The property for a polynomially convex compact set  E  in      C N {C^N}       that the Siciak extremal function      Φ  E {\Phi _E}       be continuous or, equivalently, that  E  satisfy some Bernstein type inequality, is proved to be invariant under a large class of holomorphic mappings with values in      C M ( M ⩽  N ) {C^M}(M \leqslant N)       including all open holomorphic mappings. Local specifications of this result are also given.

Invariance of the 𝐿-regularity of compact sets in 𝐶^{𝑁} under holomorphic mappings