Does there exist a function that is differentiable everywhere with everywhere discontinuous partial derivatives? Does there exist a function that is differentiable everywhere with everywhere discontinuous  partial derivatives?

I just had this doubt, talking about first order partials.
It is not possible  for such function  to exist.
The differentiability of $f$ implies $f$ is continuous , therefore $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$ are both of Baire class one, which means they have a dense set of points of continuity .   

By the way, I am also curious about if there exists a function with both partial derivative exist everywhere but are continuous nowhere. Since such function is only of Baire class one and its partial derivatives are of Baire class two (not necessarily of Baire class one), it seems to have a little possibility.
Yes it does. Take $$
f\left( {x,y} \right) = \left\{ \begin{array}{l}
 x^2 \sin \left( {\frac{1}{x}} \right) + y^2 \sin \left( {\frac{1}{y}} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,xy \ne 0 \\ 
 x^2 \sin \left( {\frac{1}{x}} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \ne 0,y = 0 \\ 
 y^2 \sin \left( {\frac{1}{y}} \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0,y \ne 0 \\ 
 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0,y = 0 \\ 
 \end{array} \right..
$$ Then, $f$ is differentiable everywere however its partial derivatives are discontinuous 
$$f_x=
\left\{ \begin{array}{l}
 2x\sin \left( {\frac{1}{x}} \right) - \cos \left( {\frac{1}{x}} \right),\,\,\,\,\,\,\,\,x \ne 0 \\ 
 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0 \\ 
 \end{array} \right.
$$  and $$f_y=\left\{ \begin{array}{l}
 2y\sin \left( {\frac{1}{y}} \right) - \cos \left( {\frac{1}{y}} \right),\,\,\,\,\,\,\,\,y \ne 0 \\ 
 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y = 0 \\ 
 \end{array} \right.$$ which are discontinuous at the origin $(0,0)$. This example given in the book of B. Gelbaum and J. Olmsted, "Counterexample in Analysis", 3rd ed., page 119.