The authors study the (Berge) upper semicontinuity of a generic multifunction assigning to each parameter in a metric space a closed convex subset in Euclidean \(n\)-space. An example is the feasible set mapping associated with a parametric family of convex semi-infinite programming problems. They introduce the concept of \(\varepsilon\)-reinforced mapping, which leads to a sufficient condition for upper semicontinuity that is also necessary when the boundary of the image set at the nominal value of the parameter contains no half-lines. Using the well-known fact that a closed convex set in Euclidean \(n\)-space can be viewed as the solution set of a linear semi-infinite inequality system, an alternative form of the necessary condition is obtained.