continuous functions of several variables are Riemann summable

Theorem 1.

Continuous functions defined on compact subsets of ${\mathrm{R}}^n$ are Riemann integrable.

Proof.

Let $D\subset {ℝ}^n$ be a compact subset of ${ℝ}^n$ and let $f:D\to ℝ$ be a continuous function.Since $f$ is defined on a compact set, $f$ is uniformly continuous i.e. given $ϵ>0$ there exists $\delta >0$ such that $|x-y|\le \delta \Rightarrow |f(x)-f(y)|\le ϵ$.Let $R>0$ be large enough so that $D\subset {(-R,R)}^n$ (such an $R$ exists because $D$ is bounded).Let $P$ be a polyrectangle such that $D\subset \cup P\subset {(-R,R)}^n$ and such that every rectangle $R$ in $P$ has diameter which is less then $\delta$. So one has ${sup}_{R}f(x)-{inf}_{R}f(x)\le ϵ$ and hence

$$S^{*}(f,P)-S_{*}(f,P)\le ϵ\sum _{Q\in P}\mathrm{meas}(Q)\le ϵ\mathrm{meas}(P)\le ϵ\mathrm{meas}{[-R,R]}^n=ϵ2^n{R}^n.$$

Letting $ϵ\to 0$ one concludes that $S^{*}(f)=S_{*}(f)$.∎