Is there a formal proof of this basic integral property? This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth showing", so I was wondering if there exists a formal proof of the basic integral property:
$$\int_a^bf(x)dx=\int_a^cf(x)dx + \int_c^bf(x)dx$$
This property is absolutely crucial to calculus and it bothers me that I haven't seen any formal proof using the definition of an integral. Does anyone know of a proof?
Edit: This is all assuming that the function is continuous on the intervals [a,b], [b,c] and [a,c]
 A: At an elementary level, it is better to consider continuous functions and define the integral as a limit of sequences of left (or right) Riemann sums corresponding to regular partitions.
Let $f$ be a function continuous on $[a,b]$.
A partition of $[a,b]\,$ is a set of points $P=\{x_0,x_1,\dots,x_n\}$ with $$a=x_0<x_1<\dots<x_n=b$$ The norm of $P$ is the number $$\|P\|=\max_{1\le i\le n}\,(x_i-x_{i-1})$$ If the points are equidistant, $P$ is the n-regular partition and is denoted by $P_n\,$, so $$\|P_n\|=\frac {b-a}n$$ The number $$L(f,P)=\sum_{i=1}^n f(x_{i-1})(x_i-x_{i-1})$$ is said the left sum corresponding to $P$.
(here $L$ is for Left not for Low)
Now the comparison of two left sums.
Comparison theorem: any two left sums differ arbitrarily little from each other if the norms of the partitions to which they correspond are sufficiently small.
Formally:
For every $\varepsilon>0$ there exists $\delta>0$ such that, if $\,\|P\|<\delta\,$ and $\,\|P'\|<\delta$, then $$|L(f,P)-L(f,P')|<\varepsilon$$
($P'=\{x'_0,x'_1,\dots,x'_m\}$)
The proof is based on the (uniform) continuity of $f$.
In fact, if $$K_{ij}=[x_{i-1},x_i]\cap[x'_{j-1},x'_j]$$ then $$L(f,P)-L(f,P')=\sum_{i=1}^n \sum_{j=1}^m \,[f(x_{i-1})-f(x'_{j-1})]\,m(K_{ij})$$ where $m(K_{ij})$ is the length of the interval $K_{ij}$.
(really $K_{ij}$ can degenerate in one point or be the empty set)
Now the definition of definite integral.
The comparison theorem assures that the sequence $$S_n=L(f,P_n)$$is Cauchy so it has a limit denoted by $$\int_a^b f$$ In fact it is enough to choose $N$ such that $$\frac {b-a}N<\delta$$ to have $$|L(f,P_m)-L(,f,P_n)|<\varepsilon$$ for every $m,n \ge N$.
Finally your question.
If $a<c<b$, one also has $$S'_n=L(f,P'_n) \to \int_a^c f$$ and $$S''_n=L(f,P''_n) \to \int_c^b f$$ where $P'_n$ and $P''_n$ are the n-regular partitions of $[a,c]$ and $[c,b]$ respectively.
Note that $S'_n+S''_n$ is the left sum corresponding to $P'_n \cup P''_n$, a partition of $[a,b]$ that is not regular unless $c$ is the midpoint of $[a,b]\;$ (so it is important that the comparison theorem is valid for any partition).
Then one gets $$|S_n- S'_n-S''_n|<\varepsilon$$ for every $n\ge N$.
In fact $\,\|P'_n \cup P''_n\|<\|P_n\|<\delta\,$ for every $n\ge N$.
So one has $$|S_n- S'_n-S''_n| \to \left|\int_a^b f-\int_a^c f-\int_c^b f \right|\le \varepsilon$$ The arbitrariness of $\varepsilon$ gives the result.
