We know for sure that for functions of several variables the existence of partial derivatives does not guarantee continuity. However, if partial derivatives exist and $f$(x) has continuous first partials in a neighborhood of some point x, then $\nabla f$(x) exist, which implies there is a vector satisfies the definition of differentiability;
$f$(x+h) $-$ $f$(x) $=\nabla f$(x) $\cdot$ h $+$ $o$(h)
Therefore $f$(x) is differentiable and continuous but that seems to contradict with our first statement, since the existence of partial derivatives guarantee differentiability, which leads to continuity. So where is my mistake or misconception?