Let a b sube r be a non-degenerate open bounded interval

Question

1. Let A ⊆ R be a set, and let f: A → R be a function. Prove that if f is monotone and injective, then f is strictly monotone.

2. Let (a, b) ⊆ R be a non-degenerate open bounded interval, and let f: (a, b) → R be a function. Suppose that f is continuous, strictly increasing and bounded. Let F:[a, b] → R be defined by

Prove that F is continuous.