True or false: a function f: R → R is continuous at a point a E R if and only if there exists a sequence (n) such that în → a and ƒ(ïn) → ƒ(a). Select one: True False

Let f A B be a function. Let X be a subset of A. (a) Verify that X ≤ f−¹(ƒ(X)). (b) Give an example of sets X CA and B, and a function f : A → B such that f−¹(f(X)) £ X. (c) Prove that if f is injective then X = f−¹(ƒ(X)).

Let n be a positive integer and let f : [0..n] → [0..n] be an injective function. Define the function g : [0..n] → Z as g(x) = n - (f(x))². Prove that is also injective.

Let f : [a, b] → [a, b] such that f(f(x)) = and f(x) # x for all x e [a, b]. Show that ƒ is discontinuous at infinitely many points. (Hint: try proving it by contradiction)

Let f : (0, 0) → R be the function with f(x) = |1 – In x|. Sketch the graph of y = f(x). Give a brief justification (one sentence each) why f is not injective and why f is not surjective. Come up with a subset A of R of your choice such that the function g : A →→ R with g(x) = |1 – In x| is injective. No justification needed. Come up with a subset B of R of your choice such that the function h : (0, ∞) → B with h(x) = |1 – In x| is surjective. No justification needed.

A function f : Z → Z is defined by f(n) = 2n + 2. (a) Determine f(E), where E is the set of even integers. (b) Determine f-1(D), where D = {6k : k E Z}.