Riemann-Darboux Integrability of Subinterval

Question

I'm studying Riemann-Darboux integration. I'm trying to prove the following rather intuitive notion for integrals. Please let me know if you find any errors in this proof, as I'm self-studying this topic.

Theorem: Suppose $f$ is Riemann-Darboux integrable on $[a,b]$. Let $c\in(a,b)$. Then, $f$ is Riemann-Darboux integrable on the intervals $[a,c]$ and $[c,b]$.

Attempted Proof: Since $f$ is Riemann-Darboux integrable on $[a,b]$, for arbitrary $\epsilon>0$, there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<\epsilon$.

Let $n_p,n_{p*}$, and $n_{p'}$ be the number of partition parts in $P$, $P^*$, and $P'$, respectively. Also, let $m_i=\inf_{x\in[x_{i-1},x_i]}f(x)$ and $M_i=\sup_{x\in[x_{i-1},x_i]}f(x)$.

Consider the partition of $[a,c]$ given by $P^*=P\cap[a,c]$. Then, $$U(f,P^*)-L(f,P^*)=\sum_{i=1}^{n_{p*}}(M_i-m_i)\Delta x_i\le\sum_{i=1}^{n_{p*}}(M_i-m_i)\Delta x_i+ \sum_{n_{p*}+1}^{n_p}(M_i-m_i)\Delta x_i=\sum_{i=1}^{n_p}(M_i-m_i)\Delta x_i=U(f,P)-L(f,P)<\epsilon$$ Therefore, $f$ is integrable on $[a,c]$.

Next, consider the partition $[a,b]$ given by $P'=P\cap[c,b]$. Then,

$$U(f,P')-L(f,P')=\sum_{i=n_{p*}+1}^{n_{p'}}(M_i-m_i)\Delta x_i\le\sum_{i=1}^{n_{p*}}(M_i-m_i)\Delta x_i+ \sum_{n_{p*}+1}^{n_p}(M_i-m_i)\Delta x_i=\sum_{i=1}^{n_p}(M_i-m_i)\Delta x_i=U(f,P)-L(f,P)<\epsilon$$ Therefore, $f$ is integrable on $[c,b]$. $\square$

Any and all feedback, or alternative proofs are appreciated. I love to see different arguments to expand my skill set.

Answer 1

This is essentially fine, although you are tacitly assuming that $c$ itself is a member of the partition, $P$. This isn't a big deal though - a $P$ such that $$U(P,f)-L(P,f)<\epsilon$$ is guaranteed by integrability, and you can always just add $c$ to this partition if it is not already there. Namely, if $P_{c}$ is this new partition, called a refinement of $P$, one must have $$U(P_{c},f)-L(P_{c},f)\leq U(P,f)-L(P,f)<\epsilon$$ and you can proceed as you have done in your proof.

An alternative approach is given by Lebesgue's Criterion for Riemann Integrability, which states that Riemann/Darboux Integrability is equivalent to boundedness plus continuity up to a null set (see Wikipedia for a good explanation of null sets). Since your functions is integrable on $[a,b]$, it is bounded and continuous up to a null set on $[a,b]$, and hence, on $[a,c]$ and $[c,b]$ as well.