Exercise · ZW-INTEGRABILITY-OF-CONTINUOUS-FUNCTIONS-001 · ★☆☆ Basic
Which of these functions is integrable on \([0,1]\) because it is continuous on the closed interval?
Choose one option.
Show answer
- C. \(f(x)=x^2\)
Explanation
\(x^2\) is continuous on \([0,1]\), so it is integrable. The other three are not continuous; the \(1/x\) one is also unbounded and the other two (versions of the Dirichlet function) are not integrable.
Review the concept: Integrability of continuous functions
Result stating that every function \(f\) continuous on a closed, bounded interval \([a,b]\) is Riemann integrable there.
\[f\text{ continuous on }[a,b]\ \Rightarrow\ \int_a^b f(x)\,dx\text{ exists}\]
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