Question

55. Show that a monotone function on an open interval is continuous if and only if its image is an interval. 56. Let f be a real-valued function defined on R. Show that the set of points at which f is continuous is a Gs set.

Answer 1

55.   Let I be the (open interval) domain of f.

Pick y₁, y₂ ∈ f(I), without loss of generality assume y₁ < y₂. We would like to show that for all y ∈ (y₁, y₂), there exists an x ∈ I such that y = f(x). This is a direct consequence of the Intermediate Value Theorem: f is a continuous function on the closed interval [x₁, x₂] (with y₁ = f(x₁), y₂ = f(x₂)), so there exists an x ∈ (x₁, x₂) such that f(x) = y

conversely, Suppose that f : I → J is a monotone function on intervals I, J ⊂ R. Constant functions are automatically continuous, hence without loss of generality we may assume f is strictly increasing. Suppose that U ⊂ J is an open set, we would like to show f⁻¹(U) ⊂ I is open. For each f(x) ∈ U, there exists an ε such that (f(x)−ε, f(x)+ε) ⊂ U. Since f strictly increasing, there exist x₁, x₂ ∈ f⁻¹(U) such that f(x₁) = f(x)−ε and f(x₂) = f(x) + ε with x ∈ (x₁, x₂) ⊂ f _⁻¹(U). Hence f ⁻¹(U) is open. hence, f is continuous.