Exercise · ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-001 · ★☆☆ Basic
Let \(f:[-1,2]\to\mathbb R\), \(f(x)=x^2\). The image \(f([-1,2])\) is:
Choose one option.
Show answer
- B. \([0,4]\)
Explanation
The minimum is \(f(0)=0\) and the maximum is \(f(2)=4\). Since \([-1,2]\) is connected and \(f\) is continuous, the image is an interval: \([0,4]\).
Review the concept: Continuous image of a connected space
If \(f:X\to Y\) is a continuous function between topological spaces and \(X\) is connected, then the image \(f(X)\) is a connected subspace of \(Y\).
\[X\text{ connected},\ f\text{ continuous}\ \Longrightarrow\ f(X)\text{ connected}\]
More practice problems: continuous image of a connected space
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