GATE 2025 MA – Question 57
A nonconstant continuous f:S¹→ℝ is defined on the unit circle with usual topology. Which statements hold?
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Correct answer: (A) f maps closed sets to closed sets.; (D) For some λ, f(cosλ,sinλ)=f(−cosλ,−sinλ).
Explanation
Closed circle subsets are compact, so their images are compact and closed in ℝ, proving A. For D, define g(λ)=f(cosλ,sinλ)−f(−cosλ,−sinλ). It is continuous and g(λ+π)=−g(λ), so the intermediate value theorem gives a zero. This also refutes injectivity. Compactness makes the image bounded, refuting surjectivity. **A,D**.