Let f be continuous on [a,b] and ε>0. If |x-y|<δ implies |f(x)-f(y)|<tfracεb-a, and P has subintervals of length less than δ, what follows for U(f,P)-L(f,P)?

Exercise · ZW-INTEGRABILITY-OF-CONTINUOUS-FUNCTIONS-006 · ★★☆ Intermediate

Let \(f\) be continuous on \([a,b]\) and \(\varepsilon>0\). If \(|x-y|<\delta\) implies \(|f(x)-f(y)|<\tfrac{\varepsilon}{b-a}\), and \(P\) has subintervals of length less than \(\delta\), what follows for \(U(f,P)-L(f,P)\)?

Choose one option.

Show answer

On each subinterval \(M_i-m_i\le\tfrac{\varepsilon}{b-a}\), so \(U-L=\sum(M_i-m_i)\Delta x_i\le\tfrac{\varepsilon}{b-a}(b-a)=\varepsilon\).

Result stating that every function \(f\) continuous on a closed, bounded interval \([a,b]\) is Riemann integrable there.

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