Differentiability and continuity while partials have different conditions

The relevant things I read and will discuss are in this snapshot (from Folland Advanced calculus, and Wolfram Alpha, and this answer by zhw for an old question).

Now First question I have is without knowing $$f$$ is differentiable or continuous how do we use mean value theorem. Do We think as $$f_1$$ and $$f_2$$ are the functions of only $$x$$ and $$y$$ respectively?? Also proving this Folland uses mean value theorem Does it really necessary to use the Mean Value Theorem? Here's a picture of my proof.
1. Second question is how the functions in the picture are bounded(wolfram alpha results)? It says in question 8 that example 7 provides examples for function that satisfies this (makes $$f$$ continuous but not differentiable). The functions $$f_1$$ and $$f_2$$ are not continuous (everywhere) but they are if we take $$x$$ and $$y$$ constant respectively then they are bounded because everywhere cont functions takes bounded to bounded
2. question is In proof of zhw and mine: what happens if we remove the fact all partials are continuous? What causes it to fail in my proof? I think removing continuity causes proof to fail at where I write "by continuity". However, I don't quite see where it fails at zhw's answer. If we don't use continuity we can say "it exists" for $$f_x$$ (derivative of $$f$$ with respect to $$x$$) as well.