If a function of two variables is discontinuous at a particular point, say $(x,y)$, does this mean that the graph of that function has some hole around the point $(x,y,f(x,y))$? Is there any break in the graph at this point in certain direction?
This question arises because I have one function which is discontinuous at $(0,0)$ but all of its partial derivatives and directional derivatives exist at $(0,0)$. While calculating its partial or directional derivatives, we naturally look in a certain plane with that point and specified direction and calculate the slope of the tangent line (as you would with one variable). In my example I have all directional derivatives, which seems to imply that there is no break around $(0,0,f(0,0))$ in any direction. Then why is the function discontinuous at $(0,0)$?