Extensions of functions which preserve the continuity on the original domain

We say that a pair of topological spaces (X,Y) is good if for every A⫅X and every continuous f:A→Y there exists f̃:X→Y which extends f and is continuous at every point of A. We use this notion to characterize several classes of topological spaces, as hereditarily normal spaces, hereditarily collectionwise normal spaces, Q-spaces, and completely metrizable spaces. We also show that if X is metrizable and Y is locally compact then (X,Y) is good and we answer a question of Arhangelʹskiiʹs about weakly C-embedded subspaces. For separable metrizable spaces our classification of good pairs is almost complete, e.g., if X is uncountable Polish then (X,Y) is good if and only if Y is Polish as well. We also show that if Y is Polish and X metrizable then f̃ can be chosen to be of Baire class 1.