Let $f(x)$ be a continuous function. Let $a,b,c$ be constants, with $a < c < b$. Prove that
$\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ $(*)$
In particular, I would like to see a proof of this property that does not use the Fundamental Theorem of Calculus. I am aware that this can be easily proved using the Fundamental Theorem. However, the proof of the Fundamental Theorem of Calculus that I understand the most is the one given in James Stewart's Calculus. As it turns out, the property $(*)$ is actually used by Stewart in order to prove the Fundamental Theorem of Calculus!
As far as actually proving $(*)$ without using the Fundamental Theorem, the only thing I can think of is to use Riemann sums
$\displaystyle \int_a^b f(x)dx = \lim_{ n \to \infty } \sum_{i=1}^n f(x_i) \frac {b-a}{n}$
$\displaystyle \int_a^c f(x)dx + \int_c^b f(x)dx = \lim_{ n \to \infty } \sum_{i=1}^n f(x_i) \frac {c-a}{n} + \lim_{ m \to \infty } \sum_{j=1}^m f(x_j) \frac {b-c}{m}$
Not sure what to do next, since $x_i \neq x_j$ in general. Similarly $m = n$ is not necessarily true.
Best Answer
Hint
To see this you need to work with Riemann sums in general, not just with the case of partitions of equal length.
Consider a partition $P: a=a_0<...<a_n=c$ of $[a,c]$, and some intermediate points $x_1^*,.., x_n^*$. Consider also a partition $Q: c=b_0<...<b_m=b$ of $[c,b]$, and some intermediate points $y_1^*,.., y_m^*$.
Then the sum of the corresponding Riemann sums $$\sum_{k=1}^n f(x_k^*)(a_k-a_{k-1})+\sum_{k=1}^m f(y_k^*)(b_k-b_{k-1})$$
is a Riemann sum for $\int_a^b f(t)dt$ for the partition $$P \cup Q= a_0<a_1<...<a_n<b_1<...<b_m=b$$ and the intermediate points $x_1^*,.., x_n^*,y_1^*,.., y_m^*$.
Note here that $\| P \cup Q \| = \max\{ \|P \|, \| Q \| \}$.
Conversely, if you have a partition $P: a=a_0< a_1< ...< a_n =b$, let $k$ be the last index for which $a_k \leq c$. Then $a_{k+1}>c$.
Now, for any intermediate points $x_1,..., x_n$ show that $P': a_0<a_1<...<a_k <c$ (or $P': a_0<a_1<...<a_k =c$) and $Q': c< a_{k+1}<....<a_n=b$ are partitions of $[a,c], [c,b]$ and that $x_1^*,.., x_{k-1}^*, c$ and $x_{k}^*,.., x_n^*$ are intermediate points.
If $R$ is the corresponding Riemann sum for $P$ , and $R_1,R_2$ are the corresponding Riemann sums for $P',Q'$, show that $$|R-R_1-R_2 | < 2\|P\| M$$ where $$M= \sup\{ |f(x)| : x \in [a,b]\}$$