Part BCSIR NET December 2024on-a-finite-set-every-function-is-continuous

On a finite set every function is continuous

Let A, B be two non-empty subsets of $R$ . Let S = { $f : A \to R$ | f is continuous}, T = { $f : B \to R$ | f is continuous}, U = { $f : A \times A \to R$ | f is continuous}. Which of the following statements is true?

A.If A is finite, then there exists a bijection between S and U.✓
B.If A is finite and B = [0, 1], then there is no bijection between S and T.
C.There is no bijection between S and U for any choice of A.
D.If A ≠ B, then there is no bijection between T and U.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: Every ordered field is Archimedean

On a finite set every function is continuous

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