2. From preferences to utility Continuity V(x,y"jel,x% y, and x=limx,y=limy thenx% y, or x upper contour setsiyex: y% xj and lower contour sets yeX:x% y)are closure Proposition2: If is continuous, then exist a continuous utility function representing2.From preferences to utility • Continuity: then ,or x upper contour sets and lower contour sets are closure. • Proposition2:If is continuous, then exist a continuous utility function representing . 1 { , } , ,and lim , lim n n n n n n n n n x y x y x x y y  = → →  = = % x y % {:} y y x  X % {:} y x y  X % % %