This is a real analysis question. Suppose that (X,d) and (Y,ρ) are metric spaces. (i)The function f:X→Y is continuous if f^(−1) (E) ⊆ X is open whenever E ⊆Y is open. (ii) The function f : X → Y is continuous if for every x ∈ X and for every ε > 0, there exists a δ > 0 such that y ∈ X with d(x,y) < δ implies ρ(f(x),f(y)) < ε. The goal of this problem is to prove a third equivalent characterization. (iii) The function f : X → Y is continuous if for every x ∈ X and for every sequence {xn, n ∈ N} ⊆ X that converges to x, the sequence {f(xn), n ∈ N} ⊆ Y converges to f(x). That is, for every x ∈ X, if lim n→∞ d(xn, x) = 0, then lim n→∞ ρ(f(xn), f(x)) = 0. (a) Prove that (iii) implies (ii). (b) Prove that (i) implies (iii).
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Suppose that (X,d) and (Y,ρ) are metric spaces.
(i)The function f:X→Y is continuous if f^(−1) (E) ⊆ X is open whenever E ⊆Y is open.
(ii) The function f : X → Y is continuous if for every x ∈ X and for every ε > 0, there exists a δ > 0 such that y ∈ X with d(x,y) < δ implies ρ(f(x),f(y)) < ε.
The goal of this problem is to prove a third equivalent characterization.
(iii) The function f : X → Y is continuous if for every x ∈ X and for every sequence {xn, n ∈ N} ⊆ X that converges to x, the sequence {f(xn), n ∈ N} ⊆ Y converges to f(x). That is, for every x ∈ X, if
lim n→∞ d(xn, x) = 0, then lim n→∞ ρ(f(xn), f(x)) = 0.
(a) Prove that (iii) implies (ii).
(b) Prove that (i) implies (iii).
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