Exercise · ZW-INTEGRABILITY-OF-CONTINUOUS-FUNCTIONS-005 · ★★☆ Intermediate

In the proof for \(f\) continuous on \([a,b]\), which property lets us choose a single \(\delta\) for the whole interval?

Choose one option.

Show answer
  • B. Uniform continuity, since \([a,b]\) is closed and bounded

Explanation

On a closed, bounded interval, continuous implies uniformly continuous (Heine-Cantor); this gives a common \(\delta\) to control \(U-L\).

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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.

\[f\text{ continuous on }[a,b]\ \Rightarrow\ \int_a^b f(x)\,dx\text{ exists}\]

More practice problems: integrability of continuous functions

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