5. Does there exist a function all of whose values are equal to each other?
6. Does there exist a function whose values for integral numbers are non-integral and whose values for non-integral numbers are integral?
7. What is a function, in your opinion?
Students are accustomed to thinking of functions as “formulas.” These questions probe the depth of that misconception by asking students to consider unusual functions and functions without simple algebraic descriptions.
Justifications for 1-6 include: "Because mathematicians have decided that this is a function." "A graph of a function must be continuous." "A graph of a function cannot have two rules of correspondence. It cannot change its character.”
6. "Yes, y = 1/x."
7. "A correspondence between two sets of elements." "One factor depending on the other one." "The result of a certain rule applied to a varying number." "An operation." "It is an equation expressing a certain relation between two objects." "A mathematical expression that gives a connection between two factors." "A graph that can be described mathematically." "A collection of numbers in a certain order which can be expressed in a graph." "y = f(x)."