Assume $E ⊂ \mathbb{R}^n$ to be a closed & Jordan measurable set and let $f : E → \mathbb{R}$ be a continuous function.

Prove that $f$ is Riemann integrable over $E$.

In the answer it briefly says the result follows from the Lebesgue’s condition. So I know that the condition is " bounded function $f$ with compact support is Riemann integrable on Rn if and only if the set of its discontinuities has measure zero."

Can someone explain more in details how the result follows from the condition? it is not super clear to me.


Consider the function $g=f\cdot \chi_E$, where $$\chi_E(x)=\begin{cases}1\ \ x\in E\\0\ \ \text{otherwise}\end{cases}$$

It is now easy to see that $g$ is bounded, with support $E$ (and thus, since $E$ is closed and bounded, compact) and that the set of its discontinuities is $\partial E$. But since $E$ is Jordan measurable, $\mu(\partial E)=0$, and thus $g$ meets the requirements for the Lebesgue's condition.

Noting that $\int_{\mathbb{R}}g=\int_E f$ we obtain our assertion

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