"Differentiable", at (a, b), for a real valued function of two variables, means, there exist numbers A and B and a function such that
and such that
(Essentially that says that f(a,b)+ A(x- a)+ B(x- b) is the "best" linear approximation to f(x,y) around (a, b).)_
Now, take the limit of both sides of as (x, y) goes to (a, b). That condition on tells you that it must go to 0.