Exercise · ZW-RIEMANN-INTEGRAL-006 · ★★☆ Intermediate
Which of these functions on \([0,1]\) is not Riemann integrable?
Choose one option.
Show answer
- D. \(f(x)=1/x\) for \(x>0\), \(f(0)=0\)
Explanation
The definition requires \(f\) bounded; \(1/x\) is unbounded near \(0\), so no upper sum is finite. The others are continuous or have a single jump.
Review the concept: Riemann integral
A bounded function \(f\) on \([a,b]\) is integrable if the supremum \(L(f)\) of its lower sums and the infimum \(U(f)\) of its upper sums, over partitions \(P\), coincide; that common value is its integral.
\[L(f)=\sup_P L(f,P)=\inf_P U(f,P)=U(f)=\int_a^b f(x)\,dx\]
More practice problems: riemann integral
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