Exercise · ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-001 · ★☆☆ Basic
Let \(f:[0,1]\to\mathbb{R}\), \(f(x)=x^2+1\). What is \(f([0,1])\)?
Choose one option.
Show answer
- B. \([1,2]\)
Explanation
\(f\) is continuous and increasing on \([0,1]\), with \(f(0)=1\) and \(f(1)=2\); the image is a compact set, \([1,2]\).
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 \(Y\).
\[f(X)\ \text{compact}\]
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