I'd appreciate it if someone could lead me through the steps for this problem:
Let $f$ be defined and bounded on $[a,b]$. Define a function $g$ on $[a,b]$ by the formula $\overline{I}(\chi_{[a,x]} f)$.
- Prove $g$ is continuous on $[a,b]$.
- Suppose $f$ is continuous at $x_0$. Prove that g'(x_0) = f(x_0).
- Extend to lower integrals.
(In other words $g(x)$ is the upper integral of $f$ on $[a,x]$)