Why do partial derivatives need to be continuous in order for the function to be differentiable Why it's not enough for the partial derivatives to exist for implying differentiability of the function? 
Why is the continuity of the partial derivatives needed?
Hint : 

consider $f(x,y) = \frac{xy}{x^2+y^2}$,  $f(0,0) = 0$
To answer your first question: because they're partial, i.e., derivatives in very special directions.  Imagine, for example, a surface which is generated by a ray parallel to the $x$-$y$-axis that is rotating in a way that for the first quarter of its rotation it raises by $1$, for the next quarter it lowers by $1$, then lowers $1$ again and in its last quarter it raises again by $1$.