Okay, I'm solving this purely for personal gratification :-/
Let
so
i.e.
Now, differentiate both sides partially w.r.t x,
................(1)
Again differentiate partially w.r.t y,
................(2)
from (1) and (2)
since, x and y are arbitrary, for all x, where k is a constant (assuming total continuity and differentiability for both and
since is a constant,
since , and since ,
Thus for all
I think the best approach is to show for all rational numbers:
Since .
It simply follows now for all rational numbers
Now with an argument using continuity and the fact that every real number can be written as a infinite sum of rational numbers, it can be shown for all