Mathematics · Real Analysis
If f: (X, d1) &to; (Y, d2) is a continuous function between two metric spaces, then which of the following statements is always true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1If {xn} is a Cauchy sequence in (X, d1), then {f(xn)} must be a Cauchy sequence in (Y, d2).
- 2If {y} ⊂ Y, then the preimage {x ∈ X | f(x) = y} is not necessarily a closed set.
- 3If {xn} is a convergent sequence in (X, d1), then {f(xn)} must be a convergent sequence in (Y, d2).✓
- 4If {y} ⊂ Y, then the set {x ∈ X | f(x) ≠ y} is not necessarily an open set.
Correct answer
If {xn} is a convergent sequence in (X, d1), then {f(xn)} must be a convergent sequence in (Y, d2).
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