Suppose $f:Q\rightarrow\mathbb{R}$ is continuous, where $Q=[a_1,b_1]\times[a_2,b_2]\times\ldots\times[a_n,b_n]\subset\mathbb{R}^n$. Show that $f$ is integrable over $Q$.
Since $Q$ is closed and bounded in $\mathbb{R}^n$, it is compact. Therefore $f$ is uniformly continuous. The Riemann condition tells us that we have to prove that for any $\epsilon>0$, there exists a partition $P$ of $Q$ such that $U(f,P)-L(f,P)<\epsilon$. Equivalently, there exists a partition $P$ of $Q$ such that $$\left|\sum_Rv(R)(M_R(f)-m_R(f))\right|<\epsilon,$$ where $M_R(f)$ is the supremum of the values of $f$ inside the rectangle $R$, and $m_R(f)$ is the corresponding value for infimum. (Here, $R$ ranges over all subrectangles formed by the partition $P$.)
Best Answer
By the uniform continuity theorem, $f$ is uniformly continuous on $Q$.
Thus, given $\varepsilon>0$, there exists $\delta>0$ such that when $\mathbf x,\mathbf y \in Q$ and $||\mathbf x-\mathbf y||<\delta$, then $|f(\mathbf x)-f(\mathbf y)|<\varepsilon/v(Q)$.
Let $m$ be so large that the $||\mathbf a-\mathbf b||$ divided by $m$ is less than $\delta$ and consider the mth regular partition of $Q\,$ (defined in an obvious way).
Then, using the extreme value theorem, you have$$\left|\sum_Rv(R)(M_R(f)-m_R(f))\right|<\epsilon,$$ where $R$ ranges over all the subrectangles of the partition and $M_R(f)$ is the supremum and $m_R(f)$ the infimum of $f$ over $R$.
Therefore, by the Riemann criterion, $f$ is integrable over $Q$.