For a differentiable surjective function f : (0, 1) → (0, 1), consider the function
F : (0, 1) × (0, 1) → (0, 1) × (0, 1) given by
F(x, y) = (f(x), f(y)), x, y ∈ (0, 1). If f'(x) ≠ 0 for every x ∈ (0, 1), then which of the following statements are true?
1
F is injective.
2
f is increasing.
3
For every (x', y') ∈ (0, 1) × (0, 1), there exists a unique (x, y) ∈ (0, 1) × (0, 1) such that F(x, y) = (x', y').
4
The total derivative DF(x, y) is invertible for all (x, y) ∈ (0, 1) × (0,1).