Is any continuous function Riemann integrable on arbitrary Jordan region (domain)? Let $E\subseteq\Bbb R^n$ be a Jordan region (i.e., $\partial E$ is volume zero), and $f:E\to\Bbb R$ be a continuous function. Then is such $f$ Riemann integrable on $E$?
I remember the answer is no. However, if $f$ is bounded and continuous, is the answer turn out to be true? If so, how to prove it?
 A: This is a bit more involved than in the case of a function defined on an interval in $\mathbb{R}$.
First one defines the Riemann integral for a bounded function $f$ defined on a bounded rectangle $R \subset \mathbb{R}^n.$  The development is analogous to that for a function of a single variable.  It can be shown that if $f$ is continuous everywhere except possibly on a set of content (or measure) zero, then the integral exists.
It is important to know that the boundedness of $f$ is not just a part of the definition for the Riemann integral added for convenience, but, in fact, a necessary condition. Given the definition of the integral as a limit of Riemann sums as partitions are refined, such an integral will fail to exist if $f$ is unbounded.
Turning to the general Jordan region $E$, the integral must first be defined in the framework of rectangular regions. This is accomplished by introducing the characteristic function $\chi_E$ where $\chi_E(x) = 1$ if $x \in E$ and $\chi_E(x) = 0$ if $x \notin E$.  Choosing any bounded rectangle $R$ containing $E$, the integral is defined as
$$\tag{*}\int_E f = \int_R f \, \chi_E$$
Of course, it must and can be shown that this integral is independent of the choice for $R$.
Finally, if $f$ is continuous on $E$ then $f \chi_E$ is continuous throughout the interior of $E$ as well as $R\setminus E$ where $f(x) \chi_E(x) = 0$. Thus $f\chi_E$ is discontinuous only at points $y \in\partial E$ where $\lim_{x \to y , x \in \text{int}(E)}\, f(x) \neq 0$. Since $E$ is a Jordan set these points are contained in a set of measure zero.   Therefore, $f \chi_E$ is Riemann integrable on $R$ and the integral on the LHS of (*) exists
