The question is most probably simple, and is related to the theorem 1.17 in Tom Apostol's calculus.
Can we prove subinterval integrability theorem: "Let $a < c < b$ be real numbers. If $\int_{a}^{b} f(x) dx$ exists, then $\int_{a}^{c} f(x) dx$ also exists"?
It is easy to prove if we know that two integrals on exist, as the theorem 1.17 implies, but I wonder if the theorem above also holds (e.g. if the integrability on a larger interval implies integrability on a subinterval). If I try to prove that the way 1.17 was proved (page 86), as follows:
$$ \int_{a}^{b} f(x) dx \iff \sup\left\{ \int_{a}^{b}s \biggm{\vert} s \leq f\right\} = \inf\left\{ \int_{a}^{b}t \biggm{\vert} f \leq t\right\} $$
Where $s$ and $t$ are step functions (more details can be found in theorem 1.9 in the Calculus book, section 1.17). By the additive property of supremum and infimum (T I.33), we have
$$\begin{align} \sup\left\{ \int_{a}^{b}s \biggm{\vert} s \leq f\right\} &= \sup\left\{ \int_{a}^{c}s \biggm{\vert} s \leq f\right\} + \sup\left\{ \int_{c}^{b}s \biggm{\vert} s \leq f\right\}\\ &= \inf\left\{ \int_{a}^{c}t \biggm{\vert} f \leq t\right\} + \inf\left\{ \int_{c}^{b}t \biggm{\vert} f \leq t\right\}\\ &= \inf\left\{ \int_{a}^{b}t \biggm{\vert} f \leq t\right\} \end{align}$$
I would be done if it would somehow follow from the above that
$$ \sup\left\{ \int_{a}^{c}s \biggm{\vert} s \leq f\right\} = \inf\left\{ \int_{a}^{c}t \biggm{\vert} f \leq t\right\} $$
But it does not, so I'm not sure how to proceed, and if it is possible to be proved, without using more advance techniques than what he covered until that point in the book.
A previous answer here mentions that the theorem holds, but I'm not sure what method would the commenter use to do that.
Thanks!
