# Continuity of the Upper and Lower Integral

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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What is $\bar{I}(\chi_{[a,x]} f)$? –  user38268 Nov 21 '11 at 11:54
The upper integral of $f(x)$ on $[a,x]$ –  totok Nov 21 '11 at 12:18