Showing $f\colon \left(-\frac{\pi}{2},\frac{\pi}{2}\right)\rightarrow(-\infty,+\infty), f(x)=x+\tan{x}$ is surjective.

Question

How to prove surjectivity of the function $f$? $$f\colon \left(-\frac{\pi}{2},\frac{\pi}{2}\right)\rightarrow (-\infty,+\infty), \quad f(x)=x+\tan{x}$$ I cannot express $x$ as the number whose function is an arbitrary real number $y$. If possible, I do not want to claim $f$ is continuous in the domain, so as to calculate the limits of the function when $x$ tends to $\pi/2$ or $-\pi/2$.