Joint continuity implies separate continuity in $\mathbb R^2$

Question

I need to prove that for a function $f: \mathbb{R^2} \rightarrow \mathbb{R}$ joint continuity at $(x_0, y_0)$ implies separate continuity. I went for an epsilon-delta style proof but have not got anywhere. Am I in the right track? I have seen posts alrerady commenting on this but Iam looking for a simpler proof or explanation.

This is what I got for the moment:

Thank you, any help is welcome.