Thanks for the help. I don't have Rudin's book but neitherway I found the proof here

http://www.maths.mq.edu.au/~wchen/ln...er/mva02-d.pdf
Now I have another question related to the same theorem.

Suppose the function:

f(x,y) ={ 1 (if x=0 or y=0)

{ 0 (otherwise)

the partial derivatives at the origin are

df/dx = 0 and df/dy = 0

so they exist and are continious.

However the function is not continious in (0,0), so it can't be differentiable there. (there can't be a tangent plane)

What did I misundertood? Because I think the theorem says that if the partial derivatives exists and they are continious then the function is differentiable there.