From what I understood:

1) The existence of the Jacobian matrix does not guarantee differentiability. It is a necessary but not sufficient condition.

2) Existence of Jacobian matrix and continuity of the partial derivatives is a sufficient but not necessary condition for differentiability.

@BalarkaSen My question now is: What is the necessary and sufficient condition for the function $f(x,y)$ to be differentiable at $(a,b)$ ?

According to what I understood the necessary and sufficient is

a) $f(a+h,b+k) = f(a,b) + T(h,k) + o[(h^2+k^2)^{1/2}]$ where $T:\Bbb R^2 \to \Bbb R$

b) $f(a+h,b+k)=f(a,b)+Ah+Bk+h\phi(h,k)+k\psi(h,k)$

where $\phi,\psi \to 0$ as $(h,k)\to (0,0)$

c)$\lim_{(h,k)\to(0,0)}\frac{f(a+h,b+k)-f(a,b)-\mathbf{A}(h,k)}{\sqrt{h^2+k^2}}=‌​0$ where $\mathbf{A}:\Bbb R^2\to \Bbb R$. $\mathbf{A}$ is the linear map which is the tangent plane at $(a,b)$

DOUBTS: We now need to show that (a),(b) and (c) are identical. Also, we need to explain why (a),(b) and (c) are the necessary and sufficient conditions. Moreover we need to explain why existence of Jacobian and continuity of partial derivatives is a condition stronger than differentiability as zhw says.

I think I can understand why a and c are equivalent conditions. Now need to show b is equivalent to them.

I think zhw is correct in saying that partial derivatives need not be continuous for differentiability at a point. Take $$f(x,y)=\begin{cases}(x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & \text{ if $(x,y) \ne (0,0)$}\\0 & \text{ if $(x,y) = (0,0)$}.\end{cases}$$ for example.