(1) Prove Proposition 3.3.1, that is prove that (given the partial function f : X -> Y), the function f_bot : X_bot -> Y_bot is continuous. (2) Prove Proposition 3.3.2, that is, prove that for each domain D the function if : B_bot x (D x D) -> D defined by if(true,(d,e)) = d if(false,(d,e)) = e and if(bot,(d,e)) = bot is continuous. (3) (Exercise 3.4.2 of Winskell): Let X and Y be sets and X_bot and Y_bot be the corresponding flat domains. Show that a function f : X_bot -> Y_bot is continuous if and only if one of (a) or (b) holds: (a) f is strict, i.e. f(bot) = bot. (b) f is constant, i.e. forall x in X . f(x) = f(bot). (4) Show that the following two definitions of the ordering between functions f,g : D -> E (see Slide 17) are equivalent. f <= g := forall d in D ( f(d) <=_E g(d) ) f <=' g := forall d1,d2 in D ( d1 <=_D d2 -> f(d1) <=_E g(d2) ) (5) Prove that for F : (D->D) -> (D->D) and g : D -> D continuous, ev(fix(F),fix(g)) = lub_{k>=0} F^k(bot')(f^k(bot)) where bot is in D and bot' is in D -> D and ev is the evaluation function of Proposition 3.2.1.