Exercise · ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-010 · ★★★ Advanced
Let \(f:[0,1]\to\mathbb{R}\) be continuous. Which statement is always true?
Choose one option.
Show answer
- C. \(f\) is bounded and attains its maximum and minimum
Explanation
\(f([0,1])\) is compact, and a compact subset of \(\mathbb{R}\) is bounded and contains its supremum and infimum. A constant function shows it need not be injective, open, or infinite.
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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