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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 $[a,x]$)

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    The upper integral of $f(x)$ on $[a,x]$2011-11-21

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