Prove discontinuity of a 1-1 function mapping from $\mathbb{R}$ to $\mathbb{R}$

Question

I was wondering what the best approach would be to demonstrate the discontinuity of a one-to-one function $g$ defined such that it maps $\mathbb{R}$ to $\mathbb{R}$ where $f(\mathbb{R}$) has a range of $[p,q)$.

Is my best bet to approach this topologically and show the inverse image $f^{-1}(\mathbb{R})$ is not open in $\mathbb{R}$?