Suppose g:R → R is continuous. Can we say whether the set S = {x : g(x) > 0} is open? closed? neither?

1. Define a function f : Z x Z → Z by f(m, n) = 2n – 4m. (a) Prove that ƒ is not one-to-one. (b) Prove that the range of f is equal to the even numbers. In other words, show rng f = {2k|k E Z}, by using the usual methods for proving an equality of sets.

3. Is there a function f: R² → R that is not continuous but f(U) is open in R for all open sets UC R²?

1. Let f (0,00)→ R be the function with f(x)= |1 - Inx. 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.