Also, let me add two other links which I read that might be helpful: this question Continuity of one partial derivative implies differentiability, in particular this answer.
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.
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
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.
question is I solved it without using Mean Value Theorem. Again, I just can't seem to understand why do we need it because if we can prove without using it then there is no need why everyone uses it then :D
I know it is a long question but they all connected Sorry for bad format I don't know how to use fancy symbols etc. Thank you for your answers already.