Mathematics · Real Analysis
Which of the following statements is necessarily true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1Let f : &mathbb;R → &mathbb;R be a continuous function. If R ⊆ image(f) is a countable set, then f-1(R) is countable.
- 2Let f : &mathbb;R → &mathbb;R be a continuous function. Then there exists an uncountable subset T ⊆ &mathbb;R such that f(T) ⊆ &mathbb;R is uncountable.
- 3Let f : &mathbb;R → &mathbb;R be a continuous and one-to-one function. If S ⊆ &mathbb;R is a countably infinite set, then f(S) is a countably infinite set.✓
- 4The set of all finite subsets of &mathbb;Z is uncountable.
Correct answer
Let f : &mathbb;R → &mathbb;R be a continuous and one-to-one function. If S ⊆ &mathbb;R is a countably infinite set, then f(S) is a countably infinite set.
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