Given an injective space \(D\) (a continuous lattice endowed with the Scott topology) and a subspace embedding \(j:X\to Y\), Dana Scott asked whether the higher-order function \([X\to D]\to [Y\to D]\) that takes a continuous map \(f:X\to D\) to its greatest extension \(\overline f:Y\to D\) is Scott continuous. Excluding the trivial case that \(D\) is a singleton, the author shows that the answer is positive if and only if the embedding \(j\) is a proper map (in the appropriate sense of proper for \(T_0\)-spaces, namely that the inverse image of a compact saturated set is compact and the lower set of the image of a closed set is again closed). The preceding result leads to a more detailed study of proper embeddings and properly embedded subspaces. It is shown that the properly embedded sober subspaces of injective spaces are precisely the stably locally compact spaces. The injective spaces with respect to proper embeddings are shown to be the algebras of the upper power space monad (consisting of the compact saturated subsets of a space) on the category of sober spaces, which in turn coincide with the retracts of the upper power spaces of sober spaces. In the full category of locally compact sober spaces resp. stably locally compact spaces, these are the continuous meet-semilattices resp. the continuous lattices. The preceding characterization of injective spaces over proper embeddings is derived as a special case of a more general categorical result on injective objects in poset-enriched categories. The results also hold for injective spaces over dense proper subspace embeddings (continuous Scott domains). Moreover, it is shown that every sober space has a smallest proper dense sober subspace, called the support, which always contains the subspace of maximal points. In the stably locally compact case, it is the subspace of maximal points if and only if that subspace is compact.