Mappings of Baire spaces into function spaces and Kadeč renorming

Assuming that there exists in the unit interval 0, 1] a coanalytic set of continuum cardinality without any perfect subset, we show the existence of a scattered compact Hausdorff spaceK with the following properties: (i) For each continuous mapf on a Baire spaceB into (C(K), pointwise), the set of points of continuity of the mapf: B → (C(K), norm) is a denseG _δ subset ofB, and (ii)C(K) does not admit a Kadeč norm that is equivalent to the supremum norm. This answers the question of Deville, Godefroy and Haydon under the set theoretic assumption stated above.