If you want other people to evaluate your work, you should try to make it readable.
You should try to make your statements clear.
For one thing, it was not at all clear whether you were claiming that you had a proof that does not use continuity of the partials.
Secondly, your notation is truly atrocious. You can justify it all you want, but it is not readable at all. If you are truly interested in getting constructive feedback, I strongly suggest that you use a more human readable format.
For instance, where in the world are you using the continuity assumption?
Finally, if you are not a troll and are truly interested in receiving constructive mathematical feedback, you need to tone down a bit.
You use an extremely arrogant tone, and your responses to any criticism is acerbic. You should write as though you were talking to people in a university setting. For example, would you talk the way you write to your university professor?
Yes, I do not always do this. Yes, there are a lot of rude posters. But that doesn't mean we shouldn't try.