Consider a definite integral function $f(x)$. We know that:
$\int_{a}^{b}f(x) dx= \int_{a}^{c}f(x) dx + \int_{c}^{b}f(x) dx$
I have a couple of conceptual doubts regarding this splitting of interval for any integral.
Query 1: Integral is conceptually finding the area under the curve. So, for any function when we split the area under the curve if the above statement is true then the area of the function at the value $x=c$ has to be counted exactly once, either in $\int_{a}^{c}f(x)$ or in $\int_{c}^{b}f(x)$ because if its considered in both the sums the above equation cannot be true. So, which part of these two contains the area when $x=c$? How do we prove it? Most textbooks don't seem to address this part.
Query 2: Is the equation above valid for ALL integrals (without exception) including: 1. Improper integral with infinity as bounds 2. Discontinuous integrals 3. Non converging integrals? I presume the answer is yes but can someone please confirm (with the argument why its true). If not what are exceptions?