general topology - Is a continuous bijection function from a hausdorff space to a compact space a homeomorphism?

We know a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism.

But I am wondering what happens if we switch the domain and codomain. Is a continuous bijection function from a Hausdorff space to a compact space a homeomorphism?

What I think this is not true. Consider the example $f:\mathbb{R}\to [-\frac{\pi}{2},\frac{\pi}{2}]$ defined by $f(x)=\tan(x)$, then $f$ is a continuous bijection but $f^{-1}$ is not continuous.

Am I right? Thank you for any comments.