Exercise · ZW-INTEGRABILITY-OF-CONTINUOUS-FUNCTIONS-007 · ★★☆ Intermediate
The Dirichlet function is continuous nowhere. What can be said using the integrability theorem for continuous functions ?
Choose one option.
A Nothing: the theorem gives a sufficient, not a necessary, condition B That it is not integrable C That it is integrable D That its integral is 0 Show answer A. Nothing: the theorem gives a sufficient, not a necessary, condition← Back to the question
Explanation The theorem says “continuous implies integrable”, not the converse. Non-integrability of Dirichlet is proved separately: \(L=0\) and \(U=1\) for every partition .
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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