Exercise · ZW-PARTITION-AND-DARBOUX-SUMS-010 · ★★★ Advanced
Let \(f=1\) on the rationals and \(f=0\) on the irrationals, over \([0,1]\). For any partition \(P\), what is \(U(f,P)-L(f,P)\)?
Choose one option.
Show answer
- C. \(1\)
Explanation
Every subinterval contains both rationals and irrationals, so \(M_i=1\), \(m_i=0\), \(U=1\) and \(L=0\) always. Refining never closes the gap.
Review the concept: Partition and Darboux sums
A partition \(P\) of \([a,b]\) splits it into subintervals of length \(\Delta x_i\); the lower sum \(L\) and upper sum \(U\) of Darboux use the infimum \(m_i\) and the supremum \(M_i\) of \(f\) on each one.
\[L(f,P)=\sum_{i=1}^{n} m_i\,\Delta x_i,\qquad U(f,P)=\sum_{i=1}^{n} M_i\,\Delta x_i\]
More practice problems: partition and darboux sums
- ZW-PARTITION-AND-DARBOUX-SUMS-001 Let f(x)=x on [0,1] and P=\0,tfrac12,1\. Compute L(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-002 With the same f(x)=x and P=\0,tfrac12,1\ on [0,1], compute U(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-003 For f(x)=x^2 on [0,2] and P=\0,1,2\, compute U(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-004 Let f(x)=x^2 on [0,1] and P the partition into 4 equal parts. Compute L(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-005 Let f(x)=-x on [0,1] and P=\0,tfrac12,1\. Compute L(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-006 Let f(x)=3 on [1,4] and P any partition. What is U(f,P)?
- ZW-PARTITION-AND-DARBOUX-SUMS-007 If Q is a refinement of P (that is, P⊆ Q), which chain is correct?
- ZW-PARTITION-AND-DARBOUX-SUMS-008 For any two partitions P_1 and P_2 of [a,b], which inequality always holds?