Question

Suppose ? ⊂ R^? , ? ⊂ R^? are nonempty and open and ? : ? → R^? and ? : ? → R^? . Let ℎ : ? × ? → R ?+? be defined by ℎ(u, v) = (?(u), ?(v)). If ? is continuous at x ∈ ? and ? is continuous at y ∈ ? , then show that ℎ is continuous at (x, y) ∈ ? × ? .

Hint: Note that for any vectors z ∈ R ? and w ∈ R ? , ‖(z, w)‖ = ‖(z, 0) + (0, w)‖ ≤ ‖(z, 0)‖ + ‖(0, w)‖ = ‖z‖ + ‖w‖. And, for any vectors u ∈ R ? and v ∈ R ? , ‖(u, v)‖ ≥ max (‖u‖, ‖v‖).

Answer #1

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