continuous functions of several variables are Riemann summable

Theorem 1.

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

Proof.

Let $D\subset\mathbb{R}^{n}$ be a compact subset of $\mathbb{R}^{n}$ and let $f\colon D\to\mathbb{R}$ be a continuous function. Since $f$ is defined on a compact set, $f$ is uniformly continuous i.e. given $\epsilon>0$ there exists $\delta>0$ such that $|x-y|\leq\delta\Rightarrow|f(x)-f(y)|\leq\epsilon$ . 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 $\sup_{R}f(x)-\inf_{R}f(x)\leq\epsilon$ and hence

S^{*}(f,P)-S_{*}(f,P)\leq\epsilon\sum_{Q\in P}\mathrm{meas}(Q)\leq\epsilon% \mathrm{meas}(P)\leq\epsilon\mathrm{meas}[-R,R]^{n}=\epsilon 2^{n}R^{n}.

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

Title	continuous functions of several variables are Riemann summable
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