ABSTRACT. In this paper we consider the following general situation. Let A be a topological space (set of “parameters”), X a normed linear space, x ∈ X, and for each a ∈ A, let V_{a} be a nonempty subset of X. The parameter mapping is the (generally set-valued) mapping a ↦ P_{V_{a}}(x) on A.

In Section 2, we include some results of a general nature concerning semicontinuity properties of the parameter mapping in the case that each V_{a} is a finite-dimensional flat. The main result here is Theorem 2.5 which gives a sufficient condition under which the parameter mapping is upper semicontinuous. On the other hand, we give an example which shows that even in a two-dimensional polyhedral space, where the range {V_{a}|a ∈ A} consists of all one-dimensional subspaces, the parameter mapping is neither lower semicontinuous nor admits a continuous selection (Example 2.7). This is in striking contrast to the behavior of the metric projection itself, which in this situation is always lower semicontinuous and admits a continuous (even linear!) selection. In this connection, we should mention that the metric projection can be regarded, in a certain sense, as a special parameter mapping because of the formula P_{V} (x) = x + P_{V −{x}}(0) for each x ∈ X. From this it follows that any semicontinuity properties of the parameter mapping x ↦ P_{V −{x}}(0) are valid for the metric projection x ↦ P_{V} (x), and vice versa.