# 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 $D\subset {\mathbb{R}}^{n}$ be a compact subset of ${\mathbb{R}}^{n}$ and let $f:D\to \mathbb{R}$ be a continuous function.
Since $f$ is defined on a compact set, $f$ is uniformly continuous^{} i.e. given $\u03f5>0$ there exists $\delta >0$ such that $|x-y|\le \delta \Rightarrow |f(x)-f(y)|\le \u03f5$.
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 ${sup}_{R}f(x)-{inf}_{R}f(x)\le \u03f5$ and hence

$${S}^{*}(f,P)-{S}_{*}(f,P)\le \u03f5\sum _{Q\in P}\mathrm{meas}(Q)\le \u03f5\mathrm{meas}(P)\le \u03f5\mathrm{meas}{[-R,R]}^{n}=\u03f5{2}^{n}{R}^{n}.$$ |

Letting $\u03f5\to 0$ one concludes that ${S}^{*}(f)={S}_{*}(f)$. ∎

Title | continuous functions of several variables are Riemann summable |
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Canonical name | ContinuousFunctionsOfSeveralVariablesAreRiemannSummable |

Date of creation | 2013-03-22 15:07:56 |

Last modified on | 2013-03-22 15:07:56 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 9 |

Author | paolini (1187) |

Entry type | Theorem |

Classification | msc 26A42 |