Let A ⊆ ℝ be the closed interval [1, 3] equipped with the standard topology. Which of the following statements is correct?

Let E = [0, 1] and f: ℝ → ℝ be defined as f(x) = d(x, E), where d is the standard metric on ℝ defined as d(x, y) = |x - y|. Then which of the following options is true?

Let (X, d1) and (Y, d2) be metric spaces. If f: (X, d1) → (Y, d2) is a continuous function, then which of the following statements is NOT always true?