Take two cases. The set S of such points x k is infinite and has no limit in R n . The usefulness of this theorem can hardly be overstated - but the simplest, in our case, is that in Euclidian space, if a set X is closed and bounded, then we can claim it is compact. But X is compact. The Weierstrass Maximum Theorem We want to prove the Weierstrass Theorem, i.e. that a continuous, real-valued function over a compact set attains a maximum and a minimum.
X is compact.
A different approach must be followed. X Y be a correspondence, where X, Y R n are compact. Thus G j is closed. G j and thus G j would not be closed. Thus j is upper semicontinuous at x.