Question

Is there any rigorous proof for the differentiability of a surface on R3?

Answer 1

If the limit of a vector (x,y) which approaches (x0,y0) exists and is L,

then you can 'split' the limit into 2, both of which are approaches of a 1D point. In other words

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if **lim((x,y)->(x0,y0)) \[f(x,y)\] = L** (and exists)

then **lim(x->x0) \[lim(y->y0)  \[f(x,y)\] \] = L** (of course you can start with y, and continue with x

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Maybe you can use this to solve your problem. Just to clearify, if you use the transformation I suggested (outer limit is for x, inner for y), then while solving the inner limit, x is considered as a constant rather than a variable. After solving the inner limit, you can start solving the outer one for x