In Apostol's Calculus I, Section 2.18 (pg. 120), the author defines an indefinite integral as follows:
In this section we assume that $f$ is a function such that the integral $\int_{a}^{x}f(t)dt$ exists for each $x$ in an interval $[a,b]$. We shall keep $a$ and $f$ fixed and study this integral as a function of $x$. We denote the value of the integral by $A(x)$, so that we have $$ A(x)=\int_{a}^{x}f(t)dt $$ if $a \leq x \leq b$.
He then goes on to define a different indefinite integral of $f$, this time with lower limit $c$:
We use a different lower limit, say $c$, and define another indefinite integral $F$ by the equation $$ F(x)=\int_{c}^{x}f(t)dt $$
Here however, Apostol doesn't specify what interval $F$ is defined over, as he did with $A$. So I assume that $F$ is defined over $[c,b]$, as $A$ was defined over $[a,b]$ (is this assumption wrong..?)
Apostol then goes on to state some properties of indefinite integrals. One property in particular confuses me:
In general, if $F(x) = \int_{c}^{x}f(t)dt$ then we have $$\int_{a}^{b}f(t)dt=\int_{c}^{b}f(t)dt-\int_{c}^{a}f(t)dt=F(b)-F(a)$$
The reason why I'm confused here is because Apostol does not make explicit what possible values of $c$ this property holds for.
It seems that what he's saying here (implicitly) is that the propety holds for all $c\in\mathbb{R}$.. but, assuming $F$ is defined over $[c,b]$, then $F(a)$ is not defined for all choices of $c>a$; it is only defined for all $c\leq a$. So then shouldn't Apostol specify that $c$ must be less than or at most equal to $a$?
Is my assumption regarding the interval over which $F$ is defined wrong? Is $F$ not defined over $[c,b]$? Am I misunderstanding Apostol's definition of indefinite integrals?
$c$ is any point of $[a,b]$ and $F$ is defined on $[a,b]$. Convention: $\int_x^c=−\int_c^{x}$ if $x<c$.