Prove that for any partition of [,], (f,)=0.

Question

Let < and consider the function F:[,]→ℝ given by

F()=

   0  if  ≤  <

   1  if  = 

Prove that for any partition of [,], (F,)=0.

Hello, can anyone help me with this question? Much help needed. Thank you!

Answer 1

For this one, observe that with $b = x_n \implies Fb|[x_{n-1},b]_{\text{min}} = 0$ on $[x_{n-1}, b]$, which implies that $L(Fb,P) = 0$.