Define three functions $B_1, B_2, B_3$ as follows: $B_j(x) = 0$ if $x < 0$, $B_j(x) = 1$ if $x > O$ for $j = 1, 2, 3$; and $B_1(0) = 0, B_2(0) =1, B_3(0) = \frac{1}{2}$. Let $f$ be a bounded function on $[-1,1]$. (a) Prove that $f \in \mathscr{R}(B_1)$ if and only if $f(0+) = f(0)$ and that then $\int_{-1}^1 f dB_1 = f(0)$
define $f(x) = 7 $ if $x<0$ and $f(x) =0 $ if $x \ge 0$
define $P=\{x_0=-1,x_1,\dots,x_n=1\}$ st $0 \in (x_{k}, x_{k-1})$ for some $k$ , $U(f,P,B_1)=7$ and $L(f,P,B_1)=0$ then $f \notin \mathscr{R}(B_1)$
this is a counter example to the statement in the exercise
What is the mistake that I made ?