Dynamic programming
DeÖnition
A set-valued mapping, that is a correspondence,Φis called(^1) upper semi continuous if for everyxn!x,yn!yandyn 2 Φ(xn)
one gets thaty 2 Φ(x).This says that the
Graph(Φ)$f(x,y)jy 2 Φ(x)g
is a closed set;
(^2) lower semi continuous if for everyxn!xandy 2 Φ(x)there is a
sequenceyn 2 Φ(xn)such thatyn!y;
(^3) it is continuous if both upper- and lower semi continuous.