One can argue that it is defined on $x \in [a, b]$, since one way to define an integral is: $$ \int_c^x f(t) dt := \lim_{n \to \infty}\sum_{i=1}^{n} \frac{f(x_i)}{n} $$
where $x_i = c + \frac{(x-c)(2i-1)}{2n}$ for $i \in [n]$ (i.e. a midpoint Riemann sum)
and one can show that $\forall i \in [n]: f(x_i) \in (a, b)$ for any $x \in [a, b]$.
One can also argue that it is defined only on $x \in (a, b)$ by using the trapezoidal sum:
$$ \int_c^x f(t) dt := \lim_{n \to \infty}\sum_{i=1}^{n} \frac{f(x_i)+f(x_{i+1})}{2n} $$
where $x_i = c + \frac{(x-c)(i-1)}{n}$ for $i \in [n+1]$.
This definition fails when $x = a$ or $x = b$, since $f(x_{n+1})$ is not defined. So the integral must be defined on $x \in (a, b)$.
It is also possible to argue for $x \in (a, b]$ or $x \in [a, b)$ using left/right Riemann sums.
What interval should one use to formally define the integral function?