Answer
$f(x)$ is continuous over the interval $(-\infty, \infty).$
Work Step by Step
Using Theorem $1.11:$
$f(x)=\dfrac{g(x)}{h(x)}\to g(x)=x$ and $h(x)=x^2+x+2.$
$f(x)$ is continuous as long as both $g(x)$ and $h(x)$ are continuous and
$h(x)\ne0.$
$g(x)$ is continuous over the interval $(-\infty, \infty)$ and $h(x)$ is continuous over the interval $(-\infty, \infty)$; furthermore, $h(x)$ is never equal to zero since the determinant of the quadratic, $\Delta=(1)^2-4(1)(2)=-7\to$ no real roots.
All this shows that $f(x)$ is continuous over the interval $(-\infty, \infty).$
