My book gives the following differentiability theorem:
If the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$, then $f$ is differentiable at $(a,b)$.
If the partial derivatives $f_x$ and $f_y$ are continuous near $(a,b)$, they exist near $(a,b)$. So can we also state the differentiability theorem as:
If the partial derivatives $f_x$ and $f_y$ are continuous near $(a,b)$, then $f$ is differentiable at $(a,b)$.
Yes, it is correct. You will find a proof of the first statement in many textbboks, such as, for instance, Elementary Classical Analysis, by Jerrold Marsden and Michael J. Hoffman. Actually, the standard proof (which uses the Mean Value Theorem) is not hard.
Note that the second statement (which is easily deduced from the first one) assumes continuity at all points near $(a,b)$ and deduces from it that the function is differentiable in that area.