Stereographic projection $\mathbb{C}\mathbb{P}^1\longrightarrow\mathcal{S}^2$ between complex projective line and $2$-sphere is continuous

Question

The stereographic projection between the $2$-sphere $\mathcal{S}^2$ and the complex projective line $\mathbb{C}\mathbb{P}^1$ can be given by the following (well-defined) map $f:\mathbb{C}\mathbb{P}^1\longrightarrow\mathcal{S}^2$ defined by $f[z:w]=\left(\frac{2\text{Re}(w\overline{z})}{|w|^2+|z|^2},\frac{2\text{Im}(w\overline{z})}{|w|^2+|z|^2},\frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$. I want to show that this $f$ is a homeomorphism, but I am stuck when trying to prove its continuity. Showing that the preimage of an open subset of the sphere is an open subset op the complex projective line does not seem to be the right approach to this problem. Does someone know which approach is best here to show that $f$ is continuous? Thanks for your help!