Prove $\int_a^b f(x)\,\mathrm dx=\int_a^cf(x)\,\mathrm dx+\int_c^bf(x)\,\mathrm dx$ without the Fundamental Theorem of Calculus 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. 
 A: 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 $$M= \sup\{ |f(x)| : x \in [a,b]\}$$
A: Assuming we know additivity of integral, if we define
$$
\begin{split}
f_{ab} &= 
\begin{cases}
f(x) & x \in [a,b] \\
0 & x \not\in [a,b]
\end{cases}\\
f_{cb} &= 
\begin{cases}
f(x) & x \in [c,b] \\
0 & x \not\in [c,b]
\end{cases}\\
f_{ac} &= 
\begin{cases}
f(x) & x \in [a,c] \\
0 & x \not\in [a,c]
\end{cases}\\
\end{split}
$$
Then
$$
\begin{split}
\int_a^b f(x) \,\operatorname d x &= \int_{\mathbb R}f_{ab}(x)\, \operatorname d x  \\
\int_a^c f(x)\, \operatorname d x &= \int_{\mathbb R}f_{ac}(x) \,\operatorname d x  \\
\int_b^c f(x) \,\operatorname d x &= \int_{\mathbb R}f_{bc}(x) \,\operatorname d x  \\
\end{split}
$$
Therefore, since $
f_{ab} = f_{ac}+f_{cb} 
$, we get
$$
\begin{split}
\int_a^b f(x) \,\operatorname d x& =\int_{\mathbb R} f_{ab}(x) \,\operatorname d x \\
&= \int_{\mathbb R} \left(f_{ac}(x) + f_{cb}(x) \right)\,\operatorname d x \\
&= \int_{\mathbb R} f_{ac}(x)\, \operatorname d x +  \int_{\mathbb R}f_{cb}(x) \,\operatorname d x \\
&= \int_a^c f(x)\,\operatorname d x + \int_c^b f(x)\, \operatorname d x
\end{split}
$$
