Exercise · ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-003 · ★☆☆ Basic
What is the image of \([0,2\pi]\) under \(f(x)=\sin x\)?
Choose one option.
Show answer
- C. \([-1,1]\)
Explanation
\(\sin\) is continuous, equals \(1\) at \(\pi/2\) and \(-1\) at \(3\pi/2\), which are its maximum and minimum; the image is the compact set \([-1,1]\).
Review the concept: Continuous image of a compact space
If \(f:X\to Y\) is continuous and the space \(X\) is compact, then the image \(f(X)\) is a compact subspace of \(Y\).
\[f(X)\ \text{compact}\]
More practice problems: continuous image of a compact space
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