Let $f:[a,b]\rightarrow\mathbb{R}$. A tagged partition, $\mathcal{P}$ of $[a,b]$ is a set of ordered pairs defined as $$\mathcal{P}:=\{([x_{k−1},x_k]),t_k)\}^n_{k=0},$$ where $a=x_0<...<x_n=b$ and the "tags" $t_k∈[x_{k−1},x_k]$, where $\mathbb{P}_{[a,b]}$ is the set of all tagged partitions over $[a,b]$. $$\|\mathcal{P}\|:=\sup\{x_k-x_{k-1}|1\leq k\leq n\}$$ is the mesh of the partition. The Riemann sum of $f$ over $[a,b]$ w.r.t $\mathcal{P}$ is defined as $$S(f,\mathcal{P}):=\sum\limits^n_{k=1}f(t_k)(x_k−x_{k−1})$$ and $f$ is said to be Riemann integrable with $\int_a^bf=L$ iff $$(\forall\epsilon>0)(\exists\delta>0)(\forall \mathcal{P}\in\mathbb{P}_{[a,b]})\bigg(\|\mathcal{P}\|<\delta\Rightarrow |S(f,\mathcal{P})-L|<\epsilon\bigg)$$

$\mathfrak{R}[a,b]$ will denote set of all Riemann integrable functions on $[a,b]$.

A Cauchy integrability which is a straightforward exercise says that $f\in\mathfrak{R}[a,b]$ iff $$(\forall\epsilon>0)(\exists\delta>0)(\forall \mathcal{P}\in\mathbb{P}_{[a,b]})(\forall \mathcal{Q}\in\mathbb{P}_{[a,b]})\bigg(\|\mathcal{P}\|<\delta\wedge\|Q\|<\delta\Rightarrow |S(f,\mathcal{P})-S(f,\mathcal{Q})|<\epsilon\bigg).$$

And in fact $f\in\mathfrak{R}[a,b]$ iff for any sequence of tagged partions $\mathcal{P}_n$ on $[a,b]$ s.t. $\|\mathcal{P}_n\|\rightarrow 0$ then the sequence of Rieman sums $S(f,\mathcal{P}_n)$ is a Cauchy sequence.

Another easy equivalent condition is the following "Squeeze Theorem" for Riemann integrable functions, which states that: $f\in \mathfrak{R}[a,b]$ iff $$(\forall\epsilon>0)(\exists h_\epsilon,g_\epsilon\in\mathfrak{R}[a,b])\bigg(((\forall x)(x\in[a,b])\wedge g_\epsilon(x)\leq f(x)\leq h_\epsilon(x)))\Rightarrow\int_a^bh_\epsilon-g_\epsilon<\epsilon\bigg).$$

Using the above definition and results as well as any other relevant results, please could anyone help me show the following elementary properties of Riemann integrable functions.

(i) If $f\in\mathfrak{R}[a,b]$ then $f\in\mathfrak{R}[c,d]$ for any $a\leq c\leq d \leq b$.

(ii) If $f$ is monotonic on $[a,b]$ then $f\in\mathfrak{R}[a,b]$.

(iii) If $f\in\mathfrak{R}[a,b]$ then $f^2\in\mathfrak{R}[a,b]$.

I know that the following properties are very easy to show when one uses the Darboux definition of the Riemann integral and the upper and lower Riemann sums and in fact it is quite straight forward to show that the Darboux definition and the above Riemann sum definition are equivalent, but for completeness I want to see if these results can be shown using just the Riemann sums definition and associated results. I have pondered on these and have just hit a brick wall as it is not very amenable, so please any help and ideas will be greatly appreciated and need. Thanks in advance.

  • $\begingroup$ Here you can find the proof for (ii), here, you can find the proof for (iii) $\endgroup$ – Itay4 Apr 16 '17 at 13:20
  • $\begingroup$ Any ideas where I can find some help with (i)? Thanks $\endgroup$ – user152874 Apr 20 '17 at 13:34

Hint for (i)

You have to observe two things:

1) you can extend every partition of $[c,d]$ to a partition of $[a,b]$ with the same mesh;

being $f_1$ the restriction of $f$ to $[c,d]$

2) if you extend two tagged partitions $\mathcal {P}_1$ and $\mathcal {Q}_1$ of $[c,d]$ to tagged partitions $\mathcal {P}$ and $\mathcal {Q}$ of $[a,b]$ by the same additional points and tags, then you get $$S(f_1,\mathcal {P}_1)-S(f_1,\mathcal {Q}_1)=S(f,\mathcal {P})-S(f,\mathcal {Q})$$

So a clever use of Cauchy integrability criterion gives the proof.


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